In this paper, we propose a framework of Inexact Proximal Stochastic Second-order (IPSS) method for solving nonconvex optimization problems, whose objective function consists of an average of finitely many, possibly weakly, smooth functions and a convex but possibly nonsmooth function. At each iteration, IPSS inexactly solves a proximal subproblem constructed by using some positive definite matrix which could capture the second-order information of original problem. Proper tolerances are given for the subproblem solution in order to maintain global convergence and the desired overall complexity of the algorithm. Under mild conditions, we analyse the computational complexity related to the evaluations on the component gradient of the smooth function. We also investigate the number of evaluations of subgradient when using an iterative subgradient method to solve the subproblem. In addition, based on IPSS, we propose a linearly convergent algorithm under the proximal Polyak–Łojasiewicz condition. Finally, we extend the analysis to problems with weakly smooth function and obtain the computational complexity accordingly.

在本文中，我们提出了一个不精确的近邻随机二阶（IPSS）方法框架，用于解决非凸优化问题，其目标函数由有限个平均函数（可能是弱函数）和一个凸但可能不是光滑函数的平均值组成。在每次迭代中，IPSS不精确地解决通过使用某个正定矩阵构造的近端子问题，该正定矩阵可以捕获原始问题的二阶信息。为子问题解决方案提供了适当的容差，以保持全局收敛和算法的所需整体复杂性。在温和的条件下，我们分析了与评估平滑函数的分量梯度有关的计算复杂性。当使用迭代次梯度方法来解决子问题时，我们还将研究次梯度的评估数量。此外，基于IPSS，我们提出了在近端Polyak–Łojasiewicz条件下的线性收敛算法。最后，我们将分析扩展到具有弱平滑函数的问题，并据此获得计算复杂性。