Behavior of Gradient-Based Optimization Methods in Nonconvex Landscapes: A Dynamical Systems Perspective

Abstract: Although nonconvex optimization is NP-hard in general, local minima often provide sufficiently good solutions for many applications. Gradient descent and related methods, however, also admit saddle points as fixed points within nonconvex landscapes. Despite this, it has long been observed that gradient descent rarely converges to saddle points. Over the past decade, dynamical systems theory has provided a rigorous foundation for understanding this behavior, showing that both gradient descent and the heavy ball method almost surely avoid strict saddle points. Yet many questions remain: How long does a gradient-based method, including accelerated variants, remain near a saddle point? What parameters influence this residence time, and under what conditions exponentially fast escape is possible? What do the trajectories of these methods look like near saddle points, and can accelerated methods—where momentum alters the dynamics—exhibit similar avoidance and escape characteristics? In this talk, we examine these questions through a dynamical systems lens and present new proof techniques to better understand the behavior of gradient-based methods near saddle points. We begin by analyzing gradient descent, focusing on the structure of its trajectories and the influence of local geometry on escape. We then turn to a class of accelerated gradient methods for nonconvex problems, studying how time-varying momentum affects residence time, escape rate, and potential re-entry into saddle regions. Finally, we consider how momentum impacts the tradeoff between escape efficiency and convergence speed, offering insights into the broader performance landscape of accelerated methods.
This talk is based on joint work with Rishabh Dixit and Mert Gurbuzbalaban.

Biography: Waheed U. Bajwa is a professor and graduate director in the Department of Electrical and Computer Engineering at Rutgers University–New Brunswick, and a member of the graduate faculty in the Department of Statistics. He received his PhD in electrical engineering from the University of Wisconsin–Madison in 2009. Over the course of his career, he has held various positions in industry, as well as academic appointments at Princeton University and Duke University. His research interests span statistical signal processing, high-dimensional statistics, machine learning, inverse problems, and networked systems. In recognition of his contributions to these areas, he was elevated to the rank of IEEE Fellow in 2025. Dr. Bajwa has received numerous honors for both research and teaching, including the National Science Foundation CAREER Award, the Army Research Office Young Investigator Award, the Rutgers Presidential Outstanding Faculty Scholar Award, and the Warren I. Susman Award for Excellence in Teaching. He has co-authored award-winning papers at IEEE workshops and received recognition from the Cancer Institute of New Jersey for collaborative research. He has served in a range of editorial and leadership roles within the IEEE Signal Processing Society, including currently serving as Senior Editorial Board Member of the IEEE Signal Processing Magazine and Senior Area Editor of the IEEE Open Journal of Signal Processing. Outside the IEEE Signal Processing Society, he also serves as Associate Editor of the IEEE Transactions on Information Theory. In addition, he has contributed extensively to technical conferences and workshops and has held several elected positions on IEEE technical committees.