H\"older Gradient Descent and Adaptive Regularization Methods in Banach Spaces for First-Order Points

This paper considers optimization of smooth nonconvex functionals in smooth infinite dimensional spaces. A H"older 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 $\beta$-H"older continuous derivatives. It is shown that finding an $ϵ$-approximate first-order point requires at most $O({ϵ}^{-\frac{p+\beta }{p+\beta -1}})$ evaluations of the functional and its first $p$ derivatives.