CAREER: Embracing Local Minima and Nonsmoothness in Nonconvex Statistical Estimation: From Structures to Algorithms

优化在现代数据分析中起着至关重要的作用。复杂数据集的精确建模和鲁棒分析往往需要解决一类非光滑的优化问题，并且可能具有许多低质量的解。这些问题是具有挑战性的解决方案，并有有限的理解的解决方案返回的标准算法的属性。现有的方法通常避开这些问题或局限于其中的一小部分。这个项目的目的是大大扩大了类的问题，有效的算法存在，并为性能保证，可以获得。该项目将开发新的算法和分析工具，适用于广泛的工程和科学应用。此外，该项目还将支持一项教育计划，该计划的目标是在本科和研究生阶段连接最优化和统计学科。该项目的技术方法是基于解耦非光滑性和非凸性的一般原则，并确定局部最优解的特征结构。研究内容包括两个方面：（1）研究一类非光滑复合优化问题，并建立一个量化问题的平均情况条件和低复杂度算法收敛速度的框架;（2）考虑一类具有耦合分量的问题，刻画了局部极小点的隐结构，并利用这些结构的结果来设计和分析有效的算法，在现有的结果无法应用的设置。该项目的研究将涵盖各种重要的统计和机器学习问题。开发的技术将为统计环境中的平均情况问题提供算法性能的精细分析。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Optimization plays a crucial role in modern data analysis. Accurate modeling and robust analysis of complex datasets often require solving a class of optimization problems that are not smooth and may possess many low-quality solutions. These problems are challenging to solve, and there is limited understanding of the properties of solutions returned by standard algorithms. Existing approaches typically steer away from such problems or restrict to a small subset of them. This project aims to substantially broaden the class of problems for which efficient algorithms exist, and for which performance guarantees can be obtained. The project will develop new algorithms and analytical tools that are applicable in a broad range of engineering and science applications. Furthermore, the project will support an education plan that centers around the goal of bridging the disciplines of optimization and statistics at both undergraduate and graduate levels.The technical approaches of this project are based on the general principles of decoupling nonsmoothness and nonconvexity, and identifying the characteristic structures of locally optimal solutions. The research program consists of two main thrusts: (1) study a class of nonsmooth composite optimization problems and develop a framework for quantifying the average-case conditioning of the problems and the convergence rates of low-complexity algorithms; (2) consider a class of problems with coupled components, characterize the hidden structures of the local minima, and exploit these structural results to design and analyze efficient algorithms in settings where existing results fail to apply. The research in this project will cover a diverse set of important statistical and machine learning problems. The techniques developed will provide a refined analysis of the algorithmic performance for average-case problems in statistical settings.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.