Stochastic Optimal Control with High Dimensional Data

从登月问题到机器学习的最新发展，最优控制一直是许多突破性技术进步的核心工具。神经网络训练方面令人印象深刻的进步现在允许进一步令人兴奋的复杂和现实的应用。该项目旨在利用这些现代优化技术来构建一个几乎由数据驱动的理论，并降低最近金融危机中普遍存在的潜在灾难性模型风险。此外，随着机器学习方法成为许多行业的主导范式，这些主题的教育对国家经济至关重要。为了实现这一目标，来自各个层次的学生将被整合到这项研究中，为他们提供关于这些无所不在的计算实践的全面培训。在技术上，将深入研究几类问题，以突出和解决一般理论面临的不同困难。主要的研究将是基于深度神经网络的有效训练的最近的数值实验的一般高层分析。将分析McKean-Vlasov跳跃扩散的最优控制，以提供具体的设置。这些问题自然地设置在无限维Wasserstein型空间中，并提出了许多有趣的问题，包括构造高维但易于处理的近似。类似的问题也出现在与模型不确定性优化有关的最优运输问题，以及具有许多潜在风险因素的量化金融中的风险管理问题，它们具有广泛的适用性。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。

Optimal control has been a central tool in many groundbreaking technological advances starting with the moon-landing problem to recent developments in machine learning. Impressive advances in the training of neural networks now allow for further exciting complex and realistic applications. This project aims to leverage these modern optimization techniques to build an almost data-driven theory and to reduce the potentially catastrophic model risk that was prevalent in the recent financial crisis. Additionally, as machine learning methodology is becoming the dominant paradigm in many industries, education on these topics is vital to the economy of the nation. Towards this goal, students from all levels will be integrated into this research providing them with well-rounded training on these omnipresent computational practices. Technically, several classes of problems will be investigated in-depth to highlight and resolve different difficulties that the general theory faces. The main study will be a general high-level analysis of recent numerical experiments based on the efficient training of deep neural networks. Optimal control of McKean-Vlasov jump-diffusions will be analyzed to provide a concrete setting. These problems are naturally set in the infinite-dimensional Wasserstein-type spaces and pose many interesting questions, including the construction of high-dimensional but tractable approximations. Similar questions also arise in optimal transport problems related to optimization with model uncertainty and risk management problems in quantitative finance with many underlying risk factors, and they have broad applicability.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.