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The follow the leader (FTL) algorithm, perhaps the simplest of all onlinelearning algorithms, is known to perform well when the loss functions it isused on are convex and positively curved. In this paper we ask whether thereare other "lucky" settings when FTL achieves sublinear, "small" regret. Inparticular, we study the fundamental problem of linear prediction over anon-empty convex, compact domain. Amongst other results, we prove that thecurvature of the boundary of the domain can act as if the losses were curved:In this case, we prove that as long as the mean of the loss vectors havepositive lengths bounded away from zero, FTL enjoys a logarithmic growth rateof regret, while, e.g., for polytope domains and stochastic data it enjoysfinite expected regret. Building on a previously known meta-algorithm, we alsoget an algorithm that simultaneously enjoys the worst-case guarantees and thebound available for FTL.

Following the Leader and Fast Rates in Linear Prediction: Curved Constraint Sets and Other Regularities