  Stability is a general notion that quantifies the sensitivity of a learning
algorithm's output to small change in the training dataset (e.g. deletion or
replacement of a single training sample). Such conditions have recently been
shown to be more powerful to characterize learnability in the general learning
setting under i.i.d. samples where uniform convergence is not necessary for
learnability, but where stability is both sufficient and necessary for
learnability. We here show that similar stability conditions are also
sufficient for online learnability, i.e. whether there exists a learning
algorithm such that under any sequence of examples (potentially chosen
adversarially) produces a sequence of hypotheses that has no regret in the
limit with respect to the best hypothesis in hindsight. We introduce online
stability, a stability condition related to uniform-leave-one-out stability in
the batch setting, that is sufficient for online learnability. In particular we
show that popular classes of online learners, namely algorithms that fall in
the category of Follow-the-(Regularized)-Leader, Mirror Descent, gradient-based
methods and randomized algorithms like Weighted Majority and Hedge, are
guaranteed to have no regret if they have such online stability property. We
provide examples that suggest the existence of an algorithm with such stability
condition might in fact be necessary for online learnability. For the more
restricted binary classification setting, we establish that such stability
condition is in fact both sufficient and necessary. We also show that for a
large class of online learnable problems in the general learning setting,
namely those with a notion of sub-exponential covering, no-regret online
algorithms that have such stability condition exists.
