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Multi-Armed Bandit (MAB) framework has been successfully applied in many web applications. However, many complex real-world applications that involve multiple content recommendations cannot fit into the traditional MAB setting. To address this issue, we consider an ordered combinatorial semi-bandit problem where the learner recommends S actions from a base set of K actions, and displays the results in S (out of M) different positions. The aim is to maximize the cumulative reward with respect to the best possible subset and positions in hindsight. By the adaptation of a minimum-cost maximum-flow network, a practical algorithm based on Thompson sampling is derived for the (contextual) combinatorial problem, thus resolving the problem of computational intractability.With its potential to work with whole-page recommendation and any probabilistic models, to illustrate the effectiveness of our method, we focus on Gaussian process optimization and a contextual setting where click-through rate is predicted using logistic regression. We demonstrate the algorithms’ performance on synthetic Gaussian process problems and on large-scale news article recommendation datasets from Yahoo! Front Page Today Module.

This paper presents an algorithm, called progressive dimension growth (PDG), for the construction of linear codes with a pre-specified length and a minimum distance. A number of new linear codes over G F(5) that have been discovered via this algorithm are also presented.

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