CAREER: Modern nonconvex optimization for machine learning: foundations of geometric and scalable techniques

数学优化是机器学习（ML）和人工智能（AI）算法的核心。这里的关键挑战是决定优化什么标准，以及使用什么算法来执行优化。这些挑战是本项目的动机所在。更具体地说，该项目旨在在ML优化的三个基本主题上取得进展：（i）可以有效解决的丰富的新一类优化问题的理论基础（即，以计算上易处理的方式）;（ii）应用于机器学习中的大规模优化问题的一组算法（例如，用于加速神经网络的训练）;以及（iii）试图理解和解释神经网络在实践中成功的原因的理论。通过关注具有基础重要性的主题，该项目应该会促进各种后续研究，加深ML和AI与数学和应用科学的联系。更广泛地说，该项目也可能产生持久的社会影响，主要是因为（i）它专注于与ML和AI特别相关的优化;（2）它所连接的非传统应用领域（例如，合成生物学）;以及（3）因为研究者所处的环境会促进这种影响（即数据、系统和社会研究所（IDSS），麻省理工学院的一个跨学科研究所，其使命是推动解决社会相关问题）。最后，该项目以教育为中心，涉及学生的智力和专业发展，以及基于本文所涵盖的研究主题开发课程材料。该项目制定了一个雄心勃勃的议程，以发展几何优化，大规模非凸优化和深度神经网络的基础理论。几何优化（这是非凸优化的一个强大的新子类）的研究最初是由ML和统计学中的应用所推动的;然而，它将对所有使用优化的学科产生更广泛的影响。研究人员试图为一类严格大于通常的凸优化问题开发多项式时间优化理论，从而为从业者提供新的多项式时间工具和模型;如果成功，这项调查可能会打开整个研究和应用领域。除了几何优化，该项目还专注于大规模非凸优化以及深度学习的优化和泛化理论。在这些主题中，该项目将解决关键的理论挑战，开发可扩展的新算法，可以大大加快神经网络的训练，并取得进展，减少非凸优化的理论和现实世界的实践之间的差距。该奖项反映了NSF的法定使命，并已被认为是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的支持。

Mathematical optimization lies at the heart of machine learning (ML) and artificial intelligence (AI) algorithms. Key challenges herein are to decide what criteria to optimize, and what algorithms to use for performing the optimization. These challenges underlie the motivation for the present project. More specifically, this project seeks to make progress on three fundamental topics in optimization for ML: (i) theoretical foundations for a rich new class of optimization problems that can be solved efficiently (i.e., in a computationally tractable manner); (ii) a set of algorithms that apply to large-scale optimization problems in machine learning (e.g., for accelerating the training of neural networks); and (iii) theory that seeks to understand and explain why do neural networks succeed in practice. By focusing on topics of foundational importance, this project should spur a variety of followup research that deepends the connection of ML and AI with both mathematics and the applied sciences. More broadly, the this project may have a lasting societal impact too, primarily because of (i) its focus on optimization particularly relevant to ML and AI; (2) the non-traditional application domains it connects with (e.g., synthetic biology); and (3) because the investigator is in an environment that fosters such impact (namely, the Institute for Data, Systems, and Society (IDSS), a cross-disciplinary institute at MIT whose mission to drive solutions to problems of societal relevance). Finally, the project has an education centric focus; it involves intellectual and professional development of students, as well as development of curricular material based on the topics of research covered herein.This project lays out an ambitious agenda to develop foundational theory for geometric optimization, large-scale nonconvex optimization, and deep neural networks. The research on geometric optimization (which is a powerful new subclass of nonconvex optimization), is originally motivated by applications in ML and statistics; however, it stands to have a broader impact across all disciplines that consume optimization. The investigator seeks to develop a theory of polynomial time optimization for a class strictly larger than usual convex optimization problems, and thereby endow practitioners with new polynomial time tools and models; if successful, this investigation could open an entire subarea of research and applications. Beyond geometric optimization, the project also focuses on large-scale nonconvex optimization and on the theory of optimization and generalization for deep learning. Within these topics, the project will address key theoretical challenges, develop scalable new algorithms that could greatly speed up neural network training, and also make progress that reduces the gap between the theory and real-world practice of nonconvex optimization.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.