III: Small: A New Perspective on Grouped Variable Selection via Modern Optimization

这个项目将为分组变量选择问题研究新的统计学习方法，这些问题需要解决空间邻近性以及数据中的物理、结构和时间约束。例如，在遗传学研究中，通常知道同一遗传路径中的一组基因表现为一个组；在神经科学应用中，大脑的空间毗连区域通常表现为同质单位；在工业应用中，对于绝对协变量，具有多个水平的因素通常被视为单个单位。该项目将设计基于数学最优化的新的统计学习方法，以拓宽分组变量选择问题的有纪律的统计和计算建模的范例。这项研究还将包括对研究生的指导以及与行业合作伙伴的合作。该项目将涉及课程开发和开发供公众使用的软件。该项目将探索基于混合整数优化的计算方法，以解决分组变量选择问题。虽然基于凸松弛的方法和贪婪方法在这个问题上发挥了重要作用，但混合整数优化方法的能力和通用性在很大程度上还没有被探索过。该项目将利用过去十到十五年来数学优化领域的进展，研究这一新方向。这个项目的成功执行将创造新的工具，极大地丰富统计学家/机器学习者的工具包，使其具有原则性的计算和统计特性的可解释模型。该项目将通过使用先进的计算方法，而不是普遍使用的、计算更友好的替代方法，来调查统计性能方面的可能收益。这项研究将探索新方法与现有方法的基本联系。这一研究探索将刺激机器学习、统计学、运筹学、数学优化和应用领域的交叉领域的活动。将为所提出的方法开发软件。

This project will investigate new statistical learning methods for grouped variable selection problems that require addressing spatial proximity as well as physical, structural, and temporal constraints in data. For example, in genetic studies, it is often known that a group of genes in the same genetic pathway behaves as a group; in neuroscience applications, spatially contiguous regions of the brain often behave as homogeneous units; in industrial applications, with categorical covariates, a factor with multiple levels is often treated as a single unit. The project will design new statistical learning methods based on mathematical optimization that broadens the paradigm of disciplined statistical and computational modeling for grouped variable selection problems. The research will also involve mentoring of graduate students and collaborations with industrial partners. The project will involve curriculum development and the creation of software for public use. The project will explore computational methods based on mixed integer optimization to address the grouped variable selection problem. While convex relaxation based procedures and greedy methods have played a significant role in this problem, the power and versatility of mixed integer optimization methods have been largely unexplored. The project will investigate this new direction, leveraging the advances in this field of mathematical optimization over the past ten to fifteen years. Successful execution of this project will create new tools, significantly enriching a statistician/machine learner's toolkit of interpretable models with principled computational and statistical properties. The project will investigate possible gains in statistical performance by using advanced computational methods over popularly used, computationally friendlier alternatives. The research will explore fundamental connections of the new approaches with existing methods. The research quest will stimulate activity at the intersection of machine learning, statistics, operations research, mathematical optimization and the applied domains. Software will be developed for the methods proposed.