  This paper studies classification with an abstention option in the online
setting. In this setting, examples arrive sequentially, the learner is given a
hypothesis class $\mathcal H$, and the goal of the learner is to either predict
a label on each example or abstain, while ensuring that it does not make more
than a pre-specified number of mistakes when it does predict a label.
  Previous work on this problem has left open two main challenges. First, not
much is known about the optimality of algorithms, and in particular, about what
an optimal algorithmic strategy is for any individual hypothesis class. Second,
while the realizable case has been studied, the more realistic non-realizable
scenario is not well-understood. In this paper, we address both challenges.
First, we provide a novel measure, called the Extended Littlestone's Dimension,
which captures the number of abstentions needed to ensure a certain number of
mistakes. Second, we explore the non-realizable case, and provide upper and
lower bounds on the number of abstentions required by an algorithm to guarantee
a specified number of mistakes.
