非精确函数和随机导数的二次和三次正则化方法
Quadratic and Cubic Regularisation Methods with Inexact function and Random Derivatives for Finite-Sum Minimisation
摘要：
本文重点介绍正则化方法，使用模型的三阶搜索到二阶临界点的有限和最小化问题。所提出的变体属于[3]的框架：它采用具有足够大的前缀概率和在规定精度水平内的确定性不精确函数评估的保证精度的随机模型。在不假设无偏估计量的情况下，当分别使用二阶或三阶模型搜索一阶临界点时，预期迭代次数为$\mathcal{O}\bigl(ε_1^{-2}\bigr)$或$\mathcal{O}\bigl(ε_1^{-{3/2}}\bigr)$，当使用三阶模型搜索二阶临界点时，预期迭代次数为$\mathcal{O}\bigl(\max[ε_1^{-{3/2}},ε_2^{-3}]\bigr)$，其中$ε_j$、$j\in\{1,2\}$为$j$阶容差。这些结果与该方法的确定性对应物的最坏情况下的最优复杂度相匹配。初步的数值测试的背景下，非凸二进制分类在成像中，有和没有神经网络（ANN）的一阶最优性，提出和讨论。
英文摘要：
This paper focuses on regularisation methods using models up to the third order to search for up to second-order critical points of a finite-sum minimisation problem. The variant presented belongs to the framework of [3]: it employs random models with accuracy guaranteed with a sufficiently large prefixed probability and deterministic inexact function evaluations within a prescribed level of accuracy. Without assuming unbiased estimators, the expected number of iterations is $\mathcal{O}\bigl(ε_1^{-2}\bigr)$ or $\mathcal{O}\bigl(ε_1^{-{3/2}}\bigr)$ when searching for a first-order critical point using a second or third order model, respectively, and of $\mathcal{O}\bigl(\max[ε_1^{-{3/2}},ε_2^{-3}]\bigr)$ when seeking for second-order critical points with a third order model, in which $ε_j$, $j\in\{1,2\}$, is the $j$th-order tolerance. These results match the worst-case optimal complexity for the deterministic counterpart of the method. Preliminary numerical tests for first-order optimality in the context of nonconvex binary classification in imaging, with and without Artifical Neural Networks (ANNs), are presented and discussed.
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