The distribution of machine learning tasks on the user's devices offers several advantages for application purposes: scalability, reduction of deployment costs and privacy. We propose a basic brick, Distributed Median Elimination, which can be used to distribute the best arm identification task in various schemes. In comparison to Median Elimination run on a single player, we showed a near optimal speed-up factor. This speed-up factor is reached with a near optimal communication cost. Experiments illustrate and complete the analysis. In comparison to {\sc Median Elimination} performed on each player, according to the analysis Distributed Median Elimination shows practical improvements.
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