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Bandit Regret Scaling with the Effective Loss Range

We study how the regret guarantees of nonstochastic multi-armed bandits can be improved, if the effective range of the losses in each round is small (e.g. the maximal difference between two losses in a given round). Despite a recent impossibility result, we show how this can be made possible under certain mild additional assumptions, such as availability of rough estimates of the losses, or advance knowledge of the loss of a single, possibly unspecified arm. Along the way, we develop a novel technique which might be of independent interest, to convert any multi-armed bandit algorithm with regret depending on the loss range, to an algorithm with regret depending only on the effective range, while avoiding predictably bad arms altogether.