We consider the adaptive regularization with cubics approach for solving nonconvex optimization problems and propose a new variant based on inexact Hessian information chosen dynamically. The theoretical analysis of the proposed procedure is given. The key property of ARC framework, constituted by optimal worst-case function/derivative evaluation bounds for first- and second-order critical point, is guaranteed. Application to large-scale finite-sum minimization based on subsampled Hessian is discussed and analyzed in both a deterministic and probabilistic manner, and equipped with numerical experiments on synthetic and real datasets.

中文翻译：

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具有动态不精确Hessian信息的自适应三次正则化方法及其在有限和最小化中的应用

我们考虑采用三次方法的自适应正则化方法来解决非凸优化问题，并基于动态选择的不精确的Hessian信息提出了一种新的变体。给出了所建议程序的理论分析。保证了ARC框架的关键特性，该特性由针对一阶和二阶临界点的最佳最坏情况函数/导数评估范围构成。以确定性和概率性的方式讨论和分析了基于二次采样Hessian的大规模有限和最小化的应用，并在合成和真实数据集上进行了数值实验。