CAREER: Stochastic Nested Composition Optimization: Theory and Algorithms

这个学院早期职业发展（Career）奖的目标是为一类重要的数据驱动的随机优化问题——随机嵌套组合优化——开发基础理论和有效的计算工具。这些嵌套组合问题出现在许多领域，如风险管理、机器学习和在线决策，并且不适合经典的随机优化方法。在理论和方法方面，该项目将推进优化和数据分析。在教育方面，该项目将带来课程创新，将优化与数据分析统一起来，并为实际应用提供新的案例研究。该项目将通过让少数民族学生参与前沿研究来促进他们的发展。这项研究将产生新的算法和分析工具，对许多数据密集型应用非常有用。这些方法将在与当地医疗保健系统的合作项目中进行实证测试，目标是通过降低成本和提高服务质量来改善医疗保健服务。随机嵌套组合优化是一类涉及多期望和多层次随机变量嵌套非线性组合的新型随机优化问题。本项目将(a)建立两级和多级随机嵌套组合优化的基本复杂性理论及其推广；(b)开发有效的算法，以理论保证处理流数据；(c)调查该问题的几个特殊案例，并将结果应用于基于真实临床数据（通过与新泽西一家连锁医院合作获得）的医疗保健决策建模和优化；(d)开发创新课程和研究项目，使不同层次的学生接触到前沿技术。嵌套组合为需要在不确定情况下进行数据驱动决策和优化的应用程序提供了丰富的建模工具。一个关键的挑战是，目标不再是数据分布的线性函数，因此现有的理论和方法是不合适的。数据分布的非线性使得这个问题从根本上比大多数经典问题更加困难。克服分析的挑战需要数学规划和随机分析的结合。如果研究成功，将对扩大随机优化的研究范围做出重大贡献。从理论上讲，它将加强我们对随机优化的数学理解，并建立基本的样本复杂度界限。在方法上，它将为数据驱动优化和在线学习中的几个重要问题提供新的算法和分析工具。这些结果将在数学规划、统计学和机器学习的几个领域之间建立重要的联系。

The objective of this Faculty Early Career Development (CAREER) award is to develop foundational theory and efficient computational tools for an important class of data-driven stochastic optimization problems, stochastic nested composition optimization. These nested composition problems arise in many areas such as risk management, machine learning, and online decision making, and are not amenable to classical methods of stochastic optimization. On the theory and methodology side, the project will advance both optimization and data analytics. On the education side, the project will result in curricular innovation that integrates optimization and data analytics in a unified way and offers new case studies on practical applications. The project will promote underrepresented minority students by involving them in frontier research. The research will produce new algorithms and analysis tools that will be useful for many data-intensive applications. These methods will be empirically tested in a collaborative project with a local healthcare system, with the goal of improving healthcare delivery by reducing cost and improving quality of service. Stochastic nested composition optimization constitutes a new class of stochastic optimization problems that involve nested nonlinear composition of multiple expectations and multi-level random variables. This project will (a) establish the basic complexity theory for two-level and multi-level stochastic nested composition optimization and their generalizations; (b) develop efficient algorithms that process streaming data with theoretical guarantees; (c) investigate several special cases of the problem and apply the results to modeling and optimizing healthcare decisions based on real clinical data (obtained through the collaboration with a NJ-based hospital chain); and (d) develop innovative curricula and research projects that can bring students at various levels to frontier technology. The nested composition provides a rich modeling tool for applications that require data-driven decision-making and optimization under uncertainty. A critical challenge is that the objective is no longer a linear functional of the data distribution, and thus existing theory and methods are inappropriate. The nonlinearity with respect to the distribution of data makes the problem fundamentally more difficult than most of the classical problems. Overcoming the analytical challenge calls for an integration of mathematical programming and stochastic analysis. If successful, the research will make a substantial contribution by expanding the scope of stochastic optimization. Theoretically, it will strengthen our mathematical understanding of stochastic optimization and establish foundational sample complexity bounds. Methodologically, it will provide new algorithms and analysis tools for several important problems in data-driven optimization and online learning. The results will establish important connections among several areas in mathematical programming, statistics, and machine learning.