  This paper considers the stability of online learning algorithms and its
implications for learnability (bounded regret). We introduce a novel quantity
called {\em forward regret} that intuitively measures how good an online
learning algorithm is if it is allowed a one-step look-ahead into the future.
We show that given stability, bounded forward regret is equivalent to bounded
regret. We also show that the existence of an algorithm with bounded regret
implies the existence of a stable algorithm with bounded regret and bounded
forward regret. The equivalence results apply to general, possibly non-convex
problems. To the best of our knowledge, our analysis provides the first general
connection between stability and regret in the online setting that is not
restricted to a particular class of algorithms. Our stability-regret connection
provides a simple recipe for analyzing regret incurred by any online learning
algorithm. Using our framework, we analyze several existing online learning
algorithms as well as the "approximate" versions of algorithms like RDA that
solve an optimization problem at each iteration. Our proofs are simpler than
existing analysis for the respective algorithms, show a clear trade-off between
stability and forward regret, and provide tighter regret bounds in some cases.
Furthermore, using our recipe, we analyze "approximate" versions of several
algorithms such as follow-the-regularized-leader (FTRL) that requires solving
an optimization problem at each step.
