Paper ID sheet UCL-INMA-2018.05


On the quality of first-order approximation of functions with Hölder-continuous gradient

Guillaume Berger, P.-A. Absil, Raphaël Jungers, Yurii Nesterov
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 first-order 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.
Key words
Hölder continuous gradient; First-order Taylor approximation; Lipschitz continuous gradient; Lipschitz constant; Euclidean norms
Journal of Optimization Theory and Applications volume 185, pages 17–33 (2020)