Localization, Convexity, and Star Aggregation

• URL: http://arxiv.org/abs/2105.08866v1
• Date: Wed, 19 May 2021 00:47:59 GMT
• Title: Localization, Convexity, and Star Aggregation
• Authors: Suhas Vijaykumar
• Abstract summary: Offset Rademacher complexities have been shown to imply sharp, linear-dependent upper bounds for the square loss. We show that in the statistical setting, the offset bound can be generalized to any loss satisfying certain uniform convexity.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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, $1 < p < \infty$, 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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