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. Surprisingly, this condition is shown to also capture exponential concavity and self-concordance, uniting several apparently disparate results…