极小化一个光滑期望函数和一个非光滑凸函数之和的非光滑随机优化问题是机器学习和统计领域中最重要的模型之一。随机优化算法是求解这类问题最有效最常用的算法，其中随机一阶算法的研究已经取得了很重要的进展。为了得到收敛速度更快的数值算法，随机二阶算法的研究引起了学者们越来越多的关注和兴趣，但是目前相关研究成果还不是很多。本项目将对非光滑随机优化问题的随机二阶算法进行深入的研究，研究内容包括：半光滑子采样牛顿方法和全局化技巧相结合的随机二阶算法；基于矩阵素描技巧的半光滑牛顿-素描算法；带线搜索的随机光滑化牛顿算法。其中我们将重点研究这些算法在凸和非凸两种情况下的全局收敛性和局部收敛速度，以及这些算法的数值实现和在机器学习中的应用。预期的研究成果具有重要的理论意义和应用价值，不但可推进随机优化算法的理论研究进展，并且可为机器学习中的一些重要应用提供高效算法。

The nonsmooth stochastic optimization problem, that involving smooth expectation function and nonsmooth convex terms in the objective, frequently arises in the fields of machine learning and statistics. Stochastic first order algorithms have been taken the stage as the primary method for solving this problem due to the affordable computational costs. To accelerate the convergence, stochastic second order algorithms have gained more and more attention. However, the research on stochastic second order algorithms is quite limited in literature so far. This project will substantially study the stochastic second order algorithms for solving nonsmooth stochastic optimization problem. Specifically, the research includes: propose a hybrid stochastic second order algorithm by combining semismooth subsampled Newton steps with globalization technique; study the semismooth Newton-Sketch algorithm based on matrix sketching; develop the stochastic smoothing Newton algorithms with line search. In particular, we will analyze the global and local convergence of these algorithms in both convex and nonconvex settings, and study the numerical performance and the applications in machine learning. It is expected that the proposed algorithms are more robust and effective than the existing stochastic first order algorithms. We hope to make a significant contribution in the development of stochastic optimization algorithms and furtherly to solve some real-world problems efficiently in machine learning.