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要优化什么标准，以及使用什么算法来执行优化。这些挑战构成了本项目的动机。更具体地说，该项目寻求在ML优化中的三个基本主题上取得进展：(I)可以有效解决(即，以计算容易处理的方式)的丰富的新型优化问题的理论基础；(Ii)一套适用于机器学习中的大规模优化问题的算法(例如，用于加速神经网络的训练)；以及(Iii)寻求理解和解释为什么神经网络在实践中成功的理论。通过关注具有基础性重要性的主题，该项目应该刺激各种后续研究，加深ML和人工智能与数学和应用科学的联系。更广泛地说，这个项目也可能产生持久的社会影响，主要是因为(I)它侧重于与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)因为研究人员处于促进这种影响的环境中(即，麻省理工学院的数据、系统和社会研究所(IDSS)，这是麻省理工学院的一个跨学科研究所，其使命是推动解决具有社会相关性的问题)。最后，该项目以教育为中心；它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的课程材料的开发。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(它是非凸优化的一个强大的子类)的研究最初是由ML和统计学中的应用程序推动的；然而，它将在所有使用优化的学科中产生更广泛的影响。研究人员试图为一类严格大于通常情况的凸优化问题发展一种多项式时间优化理论，从而为实践者提供新的多项式时间工具和模型；如果成功，这项研究将开辟一个完整的研究和应用领域。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推广理论。在这些主题中，该项目将解决关键的理论挑战，开发可极大地加快神经网络训练的可扩展新算法，并取得进展，缩小非凸优化理论与现实世界实践之间的差距。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.