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We study the problem of expert advice under partial bandit feedback settingand create a sequential minimax optimal algorithm. Our algorithm works with amore general partial monitoring setting, where, in contrast to the classicalbandit feedback, the losses can be revealed in an adversarial manner. Ouralgorithm adopts a universal prediction perspective, whose performance isanalyzed with regret against a general expert selection sequence. The regret westudy is against a general competition class that covers many settings (such asthe switching or contextual experts settings) and the expert selectionsequences in the competition class are determined by the application at hand.Our regret bounds are second order bounds in terms of the sum of squared lossesand the normalized regret of our algorithm is invariant under arbitrary affinetransforms of the loss sequence. Our algorithm is truly online and does not useany preliminary information about the loss sequences.

Second Order Regret Bounds Against Generalized Expert Sequences under Partial Bandit Feedback