CRII: AF: Faster Iterative Decisions within First-order Optimization Methods

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At the heart of most user-centric algorithms today is an optimization engine trying to provide the best decision with the information observed so far in time. Examples include recommending preferred items to new consumers, computing least congested routes in a road network with changing traffic, distributing power radially over electricity grids with changing demand, and so on. All these problems have an inherent "combinatorial" structure that constrains the possible decisions one can take. This combinatorial structure becomes even more complex when these decisions are required to incorporate fairness and reduce disparate impact, such as a balanced representation of female/male scientists when a search query for "scientists" is made. These combinatorial problems are computationally challenging, necessitating rigorous yet fast algorithms to run these efficiently in real-world scenarios where decisions must be made in real-time. A large class of optimization methods prevalent in learning applications, the so-called first-order optimization methods, work by repeatedly solving perturbations of similar combinatorial subproblems. This research project will explore whether the knowledge of "how" the solution was computed to previous subproblems within first-order optimization methods can be used to speed up computations in subsequent iterations. This project has the potential of speeding up a wide range of applications whenever a constrained decision with combinatorial structure must be made in real-time as mentioned above. The interdisciplinary nature of this work, spanning first-order optimization methods and combinatorial algorithms, will benefit students helping prepare a stronger and a holistic STEM workforce with impact in a large number of applications - at both undergraduate and graduate levels.This project serves as a cornerstone for proper integration of first-order optimization methods (like Frank-Wolfe (FW), mirror descent (MD) and their variants) and the theory of combinatorial optimization with wide-ranging applications. It brings together the theory of data structures, approximation algorithms, parametric analysis in combinatorial optimization and the computational requirements of iterative first-order optimization methods to achieve amortized speeds ups in overall runtime. MD and FW variants rely only on the function value and gradient information at a single data point in each iteration and this property has rendered these methods to be used in numerous real-time machine learning applications. When making constrained combinatorial decisions as described above, these methods repeatedly compute two main subproblems: (i) linear optimization in each iteration of the FW variants, and (ii) projections or convex minimization in each iteration of the MD variants. With this award, the investigator will explore (a) identification of combinatorial primitives suitable for amortized analysis within first-order methods, (b) use of previously discovered tight cuts for iterative projections within mirror descent variants, (c) decomposition of convex problems, (d) construction of smarter warm start solutions for first-order optimization, (e) weaker but relevant approximate models and algorithms for repeated perturbed subproblems. With recent results closing the gap of possible convergence rates for first-order optimization in various settings, these questions have become of paramount importance to enable the next scale up in runtime of constrained decision-making to real-time user-centric applications.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.