Hölder Gradient Descent and Adaptive Regularization Methods in Banach Spaces for First-Order Points

Serge Gratton, Sadok Jerad, Philippe TOINT

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Abstract

This paper considers optimization of smooth nonconvex functionals in smooth
infinite dimensional spaces. A Hölder gradient descent algorithm is first
proposed for finding approximate first-order points of regularized polynomial
functionals. This method is then applied to analyze the evaluation complexity of
an adaptive regularization method which searches for approximate first-order
points of functionals with beta-Hölder continuous derivatives. It is shown
that finding an epsilon-approximate first-order point requires at most
O(epsilon^{-(p+beta)/(p+\beta-1)})
evaluations of the functional and its first p derivatives.
Original languageEnglish
Publication statusPublished - 7 Apr 2021

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