  We study a variant of decision-theoretic online learning in which the set of
experts that are available to Learner can shrink over time. This is a
restricted version of the well-studied sleeping experts problem, itself a
generalization of the fundamental game of prediction with expert advice.
Similar to many works in this direction, our benchmark is the ranking regret.
Various results suggest that achieving optimal regret in the fully adversarial
sleeping experts problem is computationally hard. This motivates our relaxation
where any expert that goes to sleep will never again wake up. We call this
setting "dying experts" and study it in two different cases: the case where the
learner knows the order in which the experts will die and the case where the
learner does not. In both cases, we provide matching upper and lower bounds on
the ranking regret in the fully adversarial setting. Furthermore, we present
new, computationally efficient algorithms that obtain our optimal upper bounds.
