We study the problem of expert advice under partial bandit feedback setting and create a sequential minimax optimal algorithm. Our algorithm works with a more general partial monitoring setting, where, in contrast to the classical bandit feedback, the losses can be revealed in an adversarial manner. Our algorithm adopts a universal prediction perspective, whose performance is analyzed with regret against a general expert selection sequence. The regret we study is against a general competition class that covers many settings (such as the switching or contextual experts settings) and the expert selection sequences 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 losses and the normalized regret of our algorithm is invariant under arbitrary affine transforms of the loss sequence. Our algorithm is truly online and does not use any preliminary information about the loss sequences.
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