CAREER: Nonconvex Optimization for Statistical Estimation and Learning: Conditioning, Dynamics, and Nonsmoothness

非凸统计估计和学习算法正在极大地提高我们从海量数据集中有效学习的能力，通过医疗保健、成像、交通和信息处理方面的新技术能力重塑社会。虽然这种学习算法在经验上取得了广泛的成功，但我们还没有找到一个连贯的数学基础，不仅可以解释它们为什么有效，它们可以解决什么任务，而且还可以解释从业者如何通过调整算法甚至任务本身来提高他们的表现。研究人员的目标是通过推进严格合理的非凸优化算法的设计，分析和部署来奠定这一基础。这项研究将创建有保证的程序，用于培训部署在政府和工业中的实用机器学习系统，以更少的数据和计算资源生成更可靠和更强大的预测模型。研究人员将把这个项目的结果纳入教育工作，包括课程开发，当地K-12推广和博士研究指导。在这个项目中，研究者设计和分析了非凸优化算法。该项目专注于简单的迭代方法，以环境形式计算数据，这是一类独特的可扩展到现代高维统计估计和学习任务的算法。该项目的首要目标是了解这些方法何时收敛到局部或全局最优值，并提供其性能的效率估计，以消耗的数据和计算资源来衡量。为了实现这一目标，调查将利用变分分析，非光滑优化，机器学习，统计和高维概率的技术。研究人员将利用这些技术来设计和装备简单，可扩展的迭代方法，用于非凸数据拟合问题，具有强大的性能保证：通用初始化策略，快速局部收敛接近最优值，以及无缝适应非光滑约束，模型，先验。这样的性能保证指导了可靠和有效的数值方法的实际实施，用于高维估计和学习。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。

Nonconvex statistical estimation and learning algorithms are dramatically improving our capacity to efficiently learn from massive datasets, reshaping society through new technological capabilities in healthcare, imaging, transportation, and information processing. Although such learning algorithms have had widespread empirical success, we have yet to find a coherent mathematical foundation that can explain not only why they work and what tasks they provably solve, but also how practitioners can improve their performance either by adjusting the algorithm or even the task itself. The investigator aims to lay this foundation by advancing the design, analysis, and deployment of rigorously justified nonconvex optimization algorithms. This research will create guaranteed procedures for training practical machine learning systems deployed in government and industry, producing more reliable and robust predictive models with fewer data and computational resources. The investigator will incorporate results from this project in education efforts, including course development, local K-12 outreach, and research mentoring of Ph.D. and undergraduate students.In this project, the investigator designs and analyzes nonconvex optimization algorithms. The project focuses on simple iterative methods that compute with data in its ambient form, a class of algorithms that are uniquely scalable to modern high-dimensional statistical estimation and learning tasks. The overarching goal of the project is to understand when these methods converge to local or global optima and to provide efficiency estimates of their performance, measured both in terms of data and computational resources consumed. To achieve this goal, the investigation will draw on the techniques of variational analysis, nonsmooth optimization, machine learning, statistics, and high-dimensional probability. The investigator will leverage these techniques to design and equip simple, scalable iterative methods for nonconvex data fitting problems with strong performance guarantees: generic initialization strategies, rapid local convergence near optima, and seamless adaptation to nonsmooth constraints, models, priors. Such performance guarantees guide the practical implementation of reliable and efficient numerical methods for high-dimensional estimation and learning.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.