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It is well known that in stochastic multi-armed bandits (MAB), the samplemean of an arm is typically not an unbiased estimator of its true mean. In thispaper, we decouple three different sources of this selection bias: adaptive\emph{sampling} of arms, adaptive \emph{stopping} of the experiment, andadaptively \emph{choosing} which arm to study. Through a new notion called``optimism'' that captures certain natural monotonic behaviors of algorithms,we provide a clean and unified analysis of how optimistic rules affect the signof the bias. The main takeaway message is that optimistic sampling induces anegative bias, but optimistic stopping and optimistic choosing both induce apositive bias. These results are derived in a general stochastic MAB setup thatis entirely agnostic to the final aim of the experiment (regret minimization orbest-arm identification or anything else). We provide examples of optimisticrules of each type, demonstrate that simulations confirm our theoreticalpredictions, and pose some natural but hard open problems.

Are sample means in multi-armed bandits positively or negatively biased?