We consider the Adversarial Multi-Armed Bandits (MAB) problem with unbounded losses, where the algorithms have no prior knowledge on the sizes of the losses. We present UMAB-NN and UMAB-G, two algorithms for non-negative and general unbounded loss respectively. For non-negative unbounded loss, UMAB-NN achieves the first adaptive and scale free regret bound without uniform exploration. Built up on that, we further develop UMAB-G that can learn from arbitrary unbounded loss. Our analysis reveals the asymmetry between positive and negative losses in the MAB problem and provide additional insights. We also accompany our theoretical findings with extensive empirical evaluations, showing that our algorithms consistently out-performs all existing algorithms that handles unbounded losses.

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