  We study a new class of online learning problems where each of the online
algorithm's actions is assigned an adversarial value, and the loss of the
algorithm at each step is a known and deterministic function of the values
assigned to its recent actions. This class includes problems where the
algorithm's loss is the minimum over the recent adversarial values, the maximum
over the recent values, or a linear combination of the recent values. We
analyze the minimax regret of this class of problems when the algorithm
receives bandit feedback, and prove that when the minimum or maximum functions
are used, the minimax regret is $\tilde \Omega(T^{2/3})$ (so called hard online
learning problems), and when a linear function is used, the minimax regret is
$\tilde O(\sqrt{T})$ (so called easy learning problems). Previously, the only
online learning problem that was known to be provably hard was the multi-armed
bandit with switching costs.
