On the Quality of First-Order Approximation of Functions with Hölder Continuous Gradient

Abstract

We show that Hölder continuity of the gradient is not only a sufficient condition, but also a necessary condition for the existence of a global upper bound on the error of the firstorder Taylor approximation. We also relate this global upper bound to the Hölder constant of the gradient. This relation is expressed as an interval, depending on the Hölder constant, in which the error of the first-order Taylor approximation is guaranteed to be. We show that, for the Lipschitz continuous case, the interval cannot be reduced. An application to the norms of quadratic forms is proposed, which allows us to derive a novel characterization of Euclidean norms.

A Hölderian backtracking method for min-max and min-min problems

“…• A byproduct of our work is a global convergence result for Hölder methods, which were earlier investigated in the literature [5,19,28,38].…”

The backtrack Hölder gradient method with application to min-max and min-min problems

“…This is done within a fairly general framework, since L is merely assumed semialgebraic while the "best response" of player II is only required to be singled-valued. A byproduct of our work is a global convergence result for Hölderian gradient methods (without backtracking) [6,20,29,39]. Our work is theoretical in essence.…”