In this paper, we propose and study opportunistic contextual bandits - a special case of contextual bandits where the exploration cost varies under different environmental conditions, such as network load or return variation in recommendations. When the exploration cost is low, so is the actual regret of pulling a sub-optimal arm (e.g., trying a suboptimal recommendation). Therefore, intuitively, we could explore more when the exploration cost is relatively low and exploit more when the exploration cost is relatively high. Inspired by this intuition, for opportunistic contextual bandits with Linear payoffs, we propose an Adaptive Upper-Confidence-Bound algorithm (AdaLinUCB) to adaptively balance the exploration-exploitation trade-off for opportunistic learning. We prove that AdaLinUCB achieves O((log T)^2) problem-dependent regret upper bound, which has a smaller coefficient than that of the traditional LinUCB algorithm. Moreover, based on both synthetic and real-world dataset, we show that AdaLinUCB significantly outperforms other contextual bandit algorithms, under large exploration cost fluctuations.
Contextual bandit algorithms have been applied to many real applications, such as display advertising @cite_19 and content recommendation @cite_2 @cite_18 . In contrast to the classic @math -arm bandit problem @cite_20 @cite_9 @cite_8 , side information called context is provided in contextual bandit problem before arm selection @cite_4 @cite_11 @cite_21 @cite_16 . The contextual bandits with linear payoffs was first introduced in @cite_4 . In @cite_2 , LinUCB algorithm is introduced based on the optimism in the face of Uncertainty" principal for linear bandits. The LinUCB algorithm and its variances are reported to be effective in real application scenarios @cite_2 @cite_6 @cite_3 @cite_1 . Compared to the classic @math -armed bandits, the contextual bandits achieves superior performance in various application scenarios @cite_13 .
@cite_18 @cite_4 @cite_8 @cite_9 @cite_21 @cite_1 @cite_6 @cite_3 @cite_19 @cite_2 @cite_16 @cite_13 @cite_20 @cite_11
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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). 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. 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.