  This paper considers the multi-armed thresholding bandit problem --
identifying all arms whose expected rewards are above a predefined threshold
via as few pulls (or rounds) as possible -- proposed by Locatelli et al. [2016]
recently. Although the proposed algorithm in Locatelli et al. [2016] achieves
the optimal round complexity in a certain sense, there still remain unsolved
issues. This paper proposes an asynchronous parallel thresholding algorithm and
its parameter-free version to improve the efficiency and the applicability. On
one hand, the proposed two algorithms use the empirical variance to guide the
pull decision at each round, and significantly improve the round complexity of
the "optimal" algorithm when all arms have bounded high order moments. The
proposed algorithms can be proven to be optimal. On the other hand, most bandit
algorithms assume that the reward can be observed immediately after the pull or
the next decision would not be made before all rewards are observed. Our
proposed asynchronous parallel algorithms allow making the choice of the next
pull with unobserved rewards from earlier pulls, which avoids such an
unrealistic assumption and significantly improves the identification process.
Our theoretical analysis justifies the effectiveness and the efficiency of
proposed asynchronous parallel algorithms.
