One of the goals of IMSI is to push the frontiers of science by providing opportunities for scholars to advance their careers by participating in IMSI activities.  Two early career scientists have recently made significant career moves as a result of the time they spent at IMSI. 

 

Gökçe Dayanıklı specializes in operations research and financial engineering.  Originally from Turkey, Gökçe got her PhD at Princeton University in 2022.  Near the end of her PhD work, she spent the Fall 2021 quarter at IMSI as a researcher in IMSI’s long program, “Distributed Solutions to Complex Societal Problems”.  In addition to giving a talk on "Optimal Incentives to Mitigate Epidemics,” during a workshop on applications of mean field games, Gökçe engaged with the other researchers in residence, established collaborations, and advanced her research.  She states: “The time I spent at IMSI as a visiting PhD student helped me connect with many academicians all around the world during the workshops. When I was on the job market, these connections provided me with very valuable advice and informed me about the open positions in their institutions that fit my profile which in turn helped me to land on my tenure track position.”  Gökçe Dayanıklı is currently an assistant professor of statistics at the University of Illinois Urbana-Champaign.

One example of the research that came out of Gökçe’s time at IMSI is the paper she authored with Mathieu Laurière entitled, “A Machine Learning Method for Stackelberg Mean Field Games.”  Their paper addresses a class of hierarchical decision-making problems in which a single leader (the “principal”) seeks to influence the behavior of a very large population of self-interested agents. Specifically, this paper studies situations where one decision-maker—such as a company, government agency, regulator, or online platform—wants to influence the behavior of a very large number of people. The interaction naturally forms a difficult bilevel optimization problem: agents respond strategically to the principal’s policy while the principal optimizes its own objective based on the resulting population behavior. The authors show that this problem can be reformulated as a single-level non-standard optimal control problem using a penalization approach, prove that the reformulation converges to the original Stackelberg mean-field Nash equilibrium, and then develop a scalable machine-learning-based solution method using feed-forward and recurrent neural networks. In other words, instead of trying to model every individual separately, Gökçe and Mathieu treat the population as a whole and ask: “What is the best policy to choose, knowing that people will react to it in their own interests?” The challenge is that predicting everyone’s response and then finding the best policy is extremely difficult computationally. To solve this they use mean field game approximation to approximate the behavior of the large population to the given policies and formulate a general form Stackelberg mean field game problem. They then develop a machine-learning approach that turns this two-step problem into a single learning task that modern neural networks can solve more efficiently. They also show mathematically that this simplification remains faithful to the original problem and demonstrate through examples that the method works well even when the problems are highly nonlinear or high dimensional. The approach could be useful for applications such as pricing products, designing incentives, regulating markets, public policy-making, and managing large online platforms.

Joe Jackson is an applied mathematician whose research focuses on applications of stochastic analysis and partial differential equations.  His current focus is the convergence problem in mean field control and mean field games.  Joe had two extended Long Program visits to IMSI that profoundly shaped his research direction and career trajectory.  His activity at IMSI was in a Long Program on Distributed Solutions to Complex Societal Problems and then the subsequent Long Program on Decision Making and Uncertainty.  Joe’s two talks and the relationships he built while at IMSI were transformative.  As he put it: “During two visits to IMSI, I was introduced to several new research areas, which helped to broaden my research program during the final years of my PhD. Ultimately, the problems which I started working on while at IMSI have become a primary focus of my research, and several of the people I met there have become close collaborators. Because of this, I feel strongly that my experiences at IMSI have helped me to become a better and more well-rounded mathematician.”  Joe Jackson is now a Dickson Instructor and NSF Postdoctoral Fellow at the University of Chicago.

As part of his research emerging from IMSI, Joe collaborated with Pierre Cardaliaguet, Samuel Daudin, and Panagiotis Souganidis( IMSI’s science advisor), on the paper “An Algebraic Convergence Rate for the Optimal Control of McKean–Vlasov Dynamics.”  Their research studies how well a simplified “mean-field” model approximates the optimal control of a very large system of interacting agents. Imagine trying to manage or influence the behavior of millions of people, vehicles, or financial accounts. Modeling every individual separately would be overwhelming, so researchers often use a statistical model that describes the crowd as a whole. This paper asks: How much accuracy do we lose by using that simplification? The authors show that as the number of individuals grows, the simplified “crowd-level” model becomes increasingly accurate, and they provide a formula that describes how quickly the approximation improves. Their result gives decision-makers confidence that solutions computed from large-scale average behavior can closely match the best possible decisions for the actual population, while requiring far less computational effort.

IMSI will continue reaching out to early career mathematical scientists and providing them with a rich scholarly and collaborative environment in which they will have opportunities to share their research and talent with other early career researchers and more senior leaders in the field.  In doing so, IMSI hopes to continue playing a catalytic role in the career advancement of mathematical scientists.