Some Topics at the Intersection of Control, Dynamics, and Learning

Dr. Eduardo Sontag

Professor, Departments of Electrical and Computer Engineering and of Bioengineering
Northeastern University

Seminar Information

Seminar Series
Dynamic Systems & Controls

Seminar Date - Time
February 21, 2025, 3:00 pm

Seminar Location
EBU2 479

Eduardo Sontag, Ph.D.

Abstract

Data-driven modeling typically involves simplifications of systems through dimensionality reduction (less variables) or through dimensionality enlargement (more variables, but simpler, perhaps linear, dynamics).  Autoencoders with narrow bottleneck layers are a typical approach to the former (allowing the discovery of dynamics taking place in a lower-dimensional manifold), while autoencoders with wide layers provide an approach to the latter, with "neurons" in these layers thought of as "observables" in Koopman representations. In the first part of this talk, I'll briefly discuss some theoretical results about each of these topics. (Joint work with M.D. Kvalheim on dimension reduction and with Z. Liu and N. Ozay on Koopman representations.)  The training of autoencoders, and more generally the solution of other optimization problems, including policy optimization in reinforcement learning, typically relies upon some variant of gradient descent. There has been much recent work in the machine learning, control, and optimization communities in the application of the Polyak-Łojasiewicz Inequality (PŁI) to such problems in order to establish exponential (a.k.a. “linear” in the local-iteration language of numerical analysis) convergence of loss functions to their minima under the gradient flow. A somewhat surprising fact is that the exponential rate, at least in the continuous-time LQR problem, vanishes for large initial conditions, resulting in a mixed globally linear / locally exponential behavior. This is in sharp contrast with the discrete-time LQR problem, where there is global exponential convergence. The gap between CT and DT behaviors motivated our work on generalizations of the PŁI condition, and the second part of the talk will address that topic. In fact, these generalizations are key to understanding the effect of errors in the estimation of the gradient. Such errors might arise from adversarial attacks, wrong evaluation by an oracle, early stopping of a simulation, inaccurate and very approximate digital twins, stochastic computations (algorithm "reproducibility"), or learning by sampling from limited data. We will suggest an input to state stability (ISS) analysis of this issue. Time permitting, we will also mention some initial results on the performance of linear feedforward networks in feedback control.  (Joint work with A.C.B. de Oliveira, L. Cui, Z.P. Jiang, and M. Siami).

Speaker Bio

Eduardo D. Sontag received his Licenciado in Mathematics at the University of Buenos Aires (1972) and a Ph.D. in Mathematics (1977) under Rudolf E. Kalman at the University of Florida. From 1977 to 2017, he was at Rutgers University, where he was a Distinguished Professor of Mathematics and a Member of the Graduate Faculty of the Departments of Computer Science and of Electrical and Computer Engineering and the Cancer Institute of NJ. He directed the undergraduate Biomathematics Interdisciplinary Major and the Center for Quantitative Biology, and was Graduate Director at the Institute for Quantitative Biomedicine. In January 2018, Dr. Sontag became a University Distinguished Professor in the Departments of Electrical and Computer Engineering and of BioEngineering at Northeastern University, where he is also affiliated with the Mathematics and the Chemical Engineering departments. Since 2006, he has been a Research Affiliate at the Laboratory for Information and Decision Systems, MIT, and since 2018 he has been a Faculty Member in the Program in Therapeutic Science at Harvard Medical School. His major current research interests lie in several areas of control and dynamical systems theory, systems molecular biology, cancer and immunology, machine learning, and computational biology. Sontag has authored over five hundred research papers and monographs and book chapters in the above areas with over 60,000 citations and an h-index of 107 (52 since 2020). He is a Fellow of various professional societies: IEEE, AMS, SIAM, and IFAC, and is also a member of SMB and BMES. He was awarded the Reid Prize in Mathematics in 2001, the 2002 Hendrik W. Bode Lecture Prize and the 2011 Control Systems Field Award from the IEEE, the 2022 Richard E. Bellman Control Heritage Award, the 2023 IFAC Triennial Award on Nonlinear Control, the 2002 Board of Trustees Award for Excellence in Research from Rutgers, and the 2005 Teacher/Scholar Award from Rutgers. In 2024, he was inducted into the American Academy of Arts and Sciences.