AF: Small: Bridging the Past and Present of Continuous Optimization for Learning

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Machine learning (ML) is being used to drive applications of artificial intelligence in many areas of science and society. Ultimately, ML problems must be distilled using statistical and mathematical techniques into problems that can be solved using computational algorithms. For the past 25 years, ML has depended heavily on the field of optimization to provide a wealth of techniques to formulate and solve ML problems. Indeed, the ML problems that optimization is called on to solve continue to grow in complexity and difficulty. Optimization tools have been applied to some of these modern ML problems, but often in ways that lack theoretical guarantees on their performance. This project aims to develop new, powerful, principled optimization approaches for solving these more complex modern ML problems. The project will study several areas of optimization that are critical to current research in ML, developing new algorithmic techniques and new theory for these areas. Priorities include the discovery of algorithms that leverage the structures that characterize various ML problems, such as sparsity, and the development of theoretical analysis to illuminate the computational performance of practical algorithms, including finite-time sample complexity bounds in the presence of nonconvexity. Three specific thrusts of the project include (i) the development of algorithms and analysis techniques for convex-concave min-max problems that take into account sparsity or regularity properties; (ii) the development of theoretically grounded algorithms for solving nonlinear programs with nonconvex functions and stochastic oracles, motivated by problems arising in constrained neural networks, problems with fairness constraints, and distributionally robust optimization; and (iii) advancing the use of optimization in reinforcement learning, such as making use of the primal-dual techniques developed in the first thrust and extending the theory of policy gradient methods to account for the inexactness that inevitably arises in practical implementations. The project's research agenda has foundations in classical optimization and more recent developments in optimization, control, learning theory, and statistics.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.