Convex and nonconvex finite-sum minimization arises in many scientific computing and machine learning applications. Recently, first-order and second-order methods where objective functions, gradients and Hessians are approximated by randomly sampling components of the sum have received great attention. We propose a new trust-region method which employs suitable approximations of the objective function, gradient and Hessian built via random subsampling techniques. The choice of the sample size is deterministic and ruled by the inexact restoration approach. We discuss local and global properties for finding approximate first- and second-order optimal points and function evaluation complexity results. Numerical experience shows that the new procedure is more efficient, in terms of overall computational cost, than the standard trust-region scheme with subsampled Hessians.

中文翻译：

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欠采样信任区域方法的不精确还原，用于有限和最小化

凸和非凸的有限和最小化出现在许多科学计算和机器学习应用中。最近，一阶和二阶方法受到关注，在该方法中，目标函数，梯度和Hessian通过对样本和进行随机采样来近似。我们提出了一种新的信赖域方法，该方法采用目标函数，梯度和通过随机子采样技术建立的Hessian的适当近似值。样本大小的选择是确定性的，并由不精确的恢复方法决定。我们讨论了局部和全局属性，以查找近似的一阶和二阶最优点以及函数评估的复杂性结果。数值经验表明，就整体计算成本而言，新程序效率更高，