CAREER: Nonconvex Optimization for Statistical Estimation and Learning: Conditioning, Dynamics, and Nonsmoothness

Nonconvex statistical estimation and learning algorithms are dramatically improving our capacity to efficiently learn from massive datasets, reshaping society through new technological capabilities in healthcare, imaging, transportation, and information processing. Although such learning algorithms have had widespread empirical success, we have yet to find a coherent mathematical foundation that can explain not only why they work and what tasks they provably solve, but also how practitioners can improve their performance either by adjusting the algorithm or even the task itself. The investigator aims to lay this foundation by advancing the design, analysis, and deployment of rigorously justified nonconvex optimization algorithms. This research will create guaranteed procedures for training practical machine learning systems deployed in government and industry, producing more reliable and robust predictive models with fewer data and computational resources. The investigator will incorporate results from this project in education efforts, including course development, local K-12 outreach, and research mentoring of Ph.D. and undergraduate students. In this project, the investigator designs and analyzes nonconvex optimization algorithms. The project focuses on simple iterative methods that compute with data in its ambient form, a class of algorithms that are uniquely scalable to modern high-dimensional statistical estimation and learning tasks. The overarching goal of the project is to understand when these methods converge to local or global optima and to provide efficiency estimates of their performance, measured both in terms of data and computational resources consumed. To achieve this goal, the investigation will draw on the techniques of variational analysis, nonsmooth optimization, machine learning, statistics, and high-dimensional probability. The investigator will leverage these techniques to design and equip simple, scalable iterative methods for nonconvex data fitting problems with strong performance guarantees: generic initialization strategies, rapid local convergence near optima, and seamless adaptation to nonsmooth constraints, models, priors. Such performance guarantees guide the practical implementation of reliable and efficient numerical methods for high-dimensional estimation and learning. 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.