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**1 - 2**of**2**### Contextual Combinatorial Cascading Bandits

"... Abstract We propose the contextual combinatorial cascading bandits, a combinatorial online learning game, where at each time step a learning agent is given a set of contextual information, then selects a list of items, and observes stochastic outcomes of a prefix in the selected items by some stopp ..."

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Abstract We propose the contextual combinatorial cascading bandits, a combinatorial online learning game, where at each time step a learning agent is given a set of contextual information, then selects a list of items, and observes stochastic outcomes of a prefix in the selected items by some stopping criterion. In online recommendation, the stopping criterion might be the first item a user selects; in network routing, the stopping criterion might be the first edge blocked in a path. We consider position discounts in the list order, so that the agent's reward is discounted depending on the position where the stopping criterion is met. We design a UCB-type algorithm, C 3 -UCB, for this problem, prove an n-step regret bound O( √ n) in the general setting, and give finer analysis for two special cases. Our work generalizes existing studies in several directions, including contextual information, position discounts, and a more general cascading bandit model. Experiments on synthetic and real datasets demonstrate the advantage of involving contextual information and position discounts.

### Combinatorial Multi-Armed Bandit with General Reward Functions

"... Abstract In this paper, we study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our fra ..."

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Abstract In this paper, we study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our framework enables a much larger class of reward functions such as the max() function and nonlinear utility functions. Existing techniques relying on accurate estimations of the means of random variables, such as the upper confidence bound (UCB) technique, do not work directly on these functions. We propose a new algorithm called stochastically dominant confidence bound (SDCB), which estimates the distributions of underlying random variables and their stochastically dominant confidence bounds. We prove that SDCB can achieve O(log T ) distribution-dependent regret andÕ( √ T ) distribution-independent regret, where T is the time horizon. We apply our results to the K-MAX problem and expected utility maximization problems. In particular, for K-MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the firstÕ( √ T ) bound on the (1− )-approximation regret of its online problem, for any > 0.