CAREER: Guaranteed Nonconvex Optimization for High-Dimensional Learning

信号处理和机器学习中的当代非凸学习方法正在彻底改变我们以自然形式处理数据的能力，从网络搜索和社交网络到医疗保健、商业和成像，为现代生活带来革命性的变化。尽管在应用中取得了广泛的经验成功，但这些学习方案缺乏明确的数学基础，无法进行严格的性能分析，也无法根据对什么、何时以及为什么有效的理解来指导系统设计。本项目旨在开发一个统一的框架来设计和分析高效的非凸优化算法。由此产生的信号估计和学习算法将部署在旨在从数据中学习可理解模型的新应用中，这反过来将允许更好的系统以更快的速度、更高的分辨率和质量获取数据。该项目的组成部分被整合到高级研究生课程中，精选结果将用于激励K-12学生追求STEM（科学，技术，工程和数学）的职业生涯。在这个项目中，研究者研究了一系列迭代算法，用于解决现代信号处理和人工智能中出现的非凸数据拟合问题，如无相成像和神经网络训练。该项目的首要目标是了解这些算法何时收敛到全局最优解，并根据关键数量（如数据样本/观测数量、关于模型的先验知识、初始化精度等）描述它们的行为和收敛速度。理论研究利用了高维概率、统计、优化和非线性动力学控制技术。理论分析指导了更可靠的学习算法的设计，这些算法可以无缝地扩展到海量数据规模，并且对现代分布式计算环境中出现的节点故障具有鲁棒性。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

Contemporary nonconvex learning approaches in signal processing and machine learning are revolutionizing our ability to process data in their natural form, bringing transformative changes to modern life ranging from web searches and social networks to healthcare, commerce, and imaging. Despite wide empirical success in applications, these learning schemes lack a clear mathematical foundation that can enable not only rigorous performance analysis but can guide system designs based on an understanding of what would work, when and why. This project aims to develop a unified framework to design and analyze efficient nonconvex optimization algorithms. The resulting signal estimation and learning algorithms will be deployed in novel applications aimed at learning understandable models from data, which in turn will allow for better systems that can acquire data faster and at higher resolution and quality. Components from this project are integrated into an advanced graduate class and select results will serve to motivate K-12 students to pursue careers in STEM (Science, Technology, Engineering and Math).In this project, the investigator studies a family of iterative algorithms for nonconvex data fitting problems that arise in modern signal processing and artificial intelligence, such as phaseless imaging and neural network training. The overarching goal of the project is to understand when these algorithms converge to globally optimal solutions and to characterize their behavior and convergence rate in terms of key quantities such as the number of data samples/observations, prior knowledge about the model, initialization accuracy, etc. The theoretical investigations utilize techniques from high-dimensional probability, statistics, optimization and nonlinear dynamics in control. The theoretical analysis guides the design of more reliable learning algorithms that can seamlessly scale to massive data sizes and are robust to node failures that arise in modern distributed computing environments.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.