Although the classical version of the Multi-Armed Bandits (MAB) framework has been applied successfully to several practical problems, in many real-world applications, the possible actions are not presented to the learner simultaneously, such as in the Internet campaign management and environmental monitoring settings. Instead, in such applications, a set of options is presented sequentially to the learner within a time span, and this process is repeated throughout a time horizon. At each time, the learner is asked whether to select the proposed option or not. We define this scenario as the Sequential Pull/No-pull Bandit setting, and we propose a meta-algorithm, namely Sequential Pull/No-pull for MAB (Seq), to adapt any classical MAB policy to better suit this setting for both the regret minimization and best-arm identification problems. By allowing the selection of multiple arms within a round, the proposed meta-algorithm gathers more information, especially in the first rounds, characterized by a high uncertainty in the arms estimate value. At the same time, the adapted algorithms provide the same theoretical guarantees as the classical policy employed. The Seq meta-algorithm was extensively tested and compared with classical MAB policies on synthetic and real-world datasets from advertising and environmental monitoring applications, highlighting its good empirical performances.
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