Localization, Convexity, and Star Aggregation
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Offset Rademacher complexities have been shown to imply sharp, data-dependent upper bounds for the square loss in a broad class of problems including improper statistical learning and online learning. We show that in the statistical setting, the offset complexity upper bound can be generalized to any loss satisfying a certain uniform convexity condition. Amazingly, this condition is shown to also capture exponential concavity and self-concordance, uniting several apparently disparate results. By a unified geometric argument, these bounds translate directly to improper learning in a non-convex class using Audibert's "star algorithm." As applications, we recover the optimal rates for proper and improper learning with the p-loss, 1 < p < ∞, closing the gap for p > 2, and show that improper variants of empirical risk minimization can attain fast rates for logistic regression and other generalized linear models.
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