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 $\epsilon$-approximate first-order point requires at most $O(\epsilon^{-\frac{p+\beta}{p+\beta-1}})$ evaluations of the functional and its first $p$ derivatives.

翻译：本文审视了平滑的非电流功能在光滑无限维度空间中的优化问题。 首先,为寻找常规化多元功能的大致第一阶点,建议使用 H\ “ 老化梯度下降算法 ” 。 然后,该方法用于分析适应性正规化方法的评估复杂性,该方法以$\beta$-H\“老式连续衍生物搜索大约第一阶功能点。 这表明, 找到一个$\ epsilon$- 近似第一阶点, 需要最多对函数及其第一个p$的衍生物进行O( \\ epsilon_\\\\\\ frac{ p\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \