We analyze the @math -armed bandit problem where the reward for each arm is a noisy realization based on an observed context under mild nonparametric assumptions. We attain tight results for top-arm identification and a sublinear regret of @math , where @math is the context dimension, for a modified UCB algorithm that is simple to implement ( @math NN-UCB). We then give global intrinsic dimension dependent and ambient dimension independent regret bounds. We also discuss recovering topological structures within the context space based on expected bandit performance and provide an extension to infinite-armed contextual bandits. Finally, we experimentally show the improvement of our algorithm over existing multi-armed bandit approaches for both simulated tasks and MNIST image classification.
Approaches to modeling the arm rewards as a function of context are most commonly parametric. One of the most popular is that of linear payoffs, studied under a minimax framework @cite_7 @cite_1 , with UCB-type algorithms @cite_2 @cite_4 @cite_9 , or with Thompson sampling @cite_8 .
@cite_4 @cite_7 @cite_8 @cite_9 @cite_1 @cite_2
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We define a general framework for a large class of combinatorial multi-armed bandit (CMAB) problems, where subsets of base arms with unknown distributions form super arms. In each round, a super arm is played and the base arms contained in the super arm are played and their outcomes are observed. We further consider the extension in which more base arms could be probabilistically triggered based on the outcomes of already triggered arms. The reward of the super arm depends on the outcomes of all 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 offline (α β)-approximation oracle that takes the means of the outcome distributions of arms and outputs a super arm that with probability β generates an α fraction of the optimal expected reward. The objective of an online learning algorithm for CMAB is to minimize (α β)-approximation regret, which is the difference in total expected reward between the αβ 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) distribution-dependent 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 of UCB1 algorithm (up to a constant factor) for the classical MAB problem, and it significantly improves the regret bound in an earlier 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 in uence maximization for viral marketing, both having nonlinear reward structures. In particular, application to social in uence maximization requires our extension on probabilistically triggered arms. 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. 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 a class of restless multi-armed bandit (RMAB) problems with unknown arm dynamics. At each time, a player chooses an arm out of @math arms to play, referred to as an active arm, and receives a random reward from a finite set of reward states. The reward state of the active arm transits according to an unknown Markovian dynamic. The reward state of passive arms (which are not chosen to play at time t) evolves according to an arbitrary unknown random process. The objective is an arm-selection policy that minimizes the regret, defined as the reward loss with respect to a player that always plays the most rewarding arm. This class of RMAB problems has been studied recently in the context of communication networks and financial investment applications. We develop a strategy that selects arms to be played in a consecutive manner in which the selection sequencing rules are adaptively updated controlled by the current sample reward means, referred to as Adaptive Sequencing Rules (ASR) algorithm. By designing judiciously the adaptive sequencing rules of the chosen arms, we show that ASR algorithm achieves a logarithmic regret order with time and a finite-sample bound on the regret is established. Although existing methods have shown a logarithmic regret order with time in this RMAB setting, the theoretical analysis presents significant improvement in the regret scaling with respect to the system parameters under ASR. Extensive simulation results support the theoretical study and demonstrate strong performance of the algorithm as compared to existing methods.