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

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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.