We study the power of different types of adaptive (nonoblivious) adversaries in the setting of prediction with expert advice, under both full-information and bandit feedback. We measure the player's performance using a new notion of regret, also known as policy regret, which better captures the adversary's adaptiveness to the player's behavior. In a setting where losses are allowed to drift, we characterize -in a nearly complete manner- the power of adaptive adversaries with bounded memories and switching costs. In particular,we show that with switching costs, the attainable rate with bandit feedback is Θ-tilde(T~(2/3)). Interestingly, this rate is significantly worse than the Θ(T~(1/2)) rate attainable with switching costs in the full-information case. Via a novel reduction from experts to bandits, we also show that a bounded memory adversary can force Θ-tilde(T~(2/3)) regret even in the full information case, proving that switching costs are easier to control than bounded memory adversaries. Our lower bounds rely on a new stochastic adversary strategy that generates loss processes with strong dependencies.
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