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This paper considers the stability of online learning algorithms and itsimplications for learnability (bounded regret). We introduce a novel quantitycalled {\em forward regret} that intuitively measures how good an onlinelearning 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 boundedregret. We also show that the existence of an algorithm with bounded regretimplies the existence of a stable algorithm with bounded regret and boundedforward regret. The equivalence results apply to general, possibly non-convexproblems. To the best of our knowledge, our analysis provides the first generalconnection between stability and regret in the online setting that is notrestricted to a particular class of algorithms. Our stability-regret connectionprovides a simple recipe for analyzing regret incurred by any online learningalgorithm. Using our framework, we analyze several existing online learningalgorithms as well as the "approximate" versions of algorithms like RDA thatsolve an optimization problem at each iteration. Our proofs are simpler thanexisting analysis for the respective algorithms, show a clear trade-off betweenstability and forward regret, and provide tighter regret bounds in some cases.Furthermore, using our recipe, we analyze "approximate" versions of severalalgorithms such as follow-the-regularized-leader (FTRL) that requires solvingan optimization problem at each step.

The Interplay Between Stability and Regret in Online Learning