  In the setting of sequential prediction of individual $\{0, 1\}$-sequences
with expert advice, we show that by allowing the learner to abstain from the
prediction by paying a cost marginally smaller than $\frac 12$ (say, $0.49$),
it is possible to achieve expected regret bounds that are independent of the
time horizon $T$. We exactly characterize the dependence on the abstention cost
$c$ and the number of experts $N$ by providing matching upper and lower bounds
of order $\frac{\log N}{1-2c}$, which is to be contrasted with the best
possible rate of $\sqrt{T\log N}$ that is available without the option to
abstain. We also discuss various extensions of our model, including a setting
where the sequence of abstention costs can change arbitrarily over time, where
we show regret bounds interpolating between the slow and the fast rates
mentioned above, under some natural assumptions on the sequence of abstention
costs.
