Mohamed-Ali Belabbas - US grants

Problems concerning the cooperation of multiple entities seeking to achieve a global objective appear in an increasingly broader variety of situations, for instance arising from the interaction of agents that were designed to act autonomously (such as cars on a highway), or from systems built from the ground up according to the networked systems paradigm (such as uav's (unmanned autonomous vehicles) or auction systems). Along with this comes an increase in the variety of tasks such systems have to accomplish. While most of the extant work in decentralized control focuses on issues relating to agents having partial information about the state of the ensemble, this new landscape underscores the need to develop new control theoretic methods that give the objective of the ensemble a prominent role. The goal of this research is to develop new results and algorithms to address this need, within the context of what is termed 'multi-agent systems with localized objectives'. The novelty of the point of view adopted here thus lies in the handling of information: beyond restrictions on the information available to the agents about the state of other agents, we integrate in the analysis restrictions on the information available about the global objective of the system as whole.

Intellectual Merit

This research project will provide novel, systematic methods for the design and analysis of multi-agent systems where each agent knows only part of the global objective of the ensemble. To this end, new connections will be established between control design and various areas of applied mathematics such as computational abstract algebra and algebraic topology.

Broader impacts

The successful completion of this research program will provide tools that enable the design of decentralized control laws with localized objectives and foster their dissemination and adoption in real-world systems such as formations of vehicles or power networks. This research plan will blend tools from different engineering disciplines and will draw on data/problems stemming from a variety of experimental origins. Its implementation will moreover involve undergraduate and graduate students from across the spectrum of engineering.

Objective: In recent years, a new paradigm for control systems, now increasingly seen as a collection of interacting autonomous agents, has emerged. A decentralized power distribution system, a formation of unmanned autonomous vehicles used for aid or even package delivery, terrestrial vehicles platoons, and auctions systems are all control systems built around this new paradigm. Similarly, progress in the biological sciences and the need to provide comprehensive and affordable healthcare have made the mapping of complex genetic, metabolic and more recently disease networks commonplace.

Intellectual merit: The research deals with a novel framework that unites ideas from graph theory, computer science and dynamical systems. The focus is on control-theoretic methods that are broad enough to address the relevant questions arising in various application settings, yet specific enough to yield implementable methods and algorithms. Specific questions include: how to characterize the minimal communication structure needed to accomplish a task (which links are necessary for the ensemble to cooperate, and which ones are redundant?), how to quantify the resistance of a network to attacks and link failures (what is a good measure of robustness?), how to construct networks accomplishing a given task (what is the most economical network design?).

Broader Impacts: The work proposed here will provide key technologies to enable the deployment and analysis of large-scale, secure and efficient multi-agent systems. Over the course of the past twenty years, the paradigm of centralized control systems has faded in favor of systems that are large-scale, heterogeneous, sometimes ad-hoc, and decentralized. Because the proposed framework directly relates the dynamics to the underlying network, its potential applications span the various subfields where networked dynamics appear, including, but not limited to, distributed optimization, the study of metabolic networks, formation control, and decentralized power distribution.

The goal of this research project is to develop a new framework for the design of secure multi-agent systems along with the tools necessary for their analysis. The rapid pace of innovation in the areas of control theory, computation, and communication is leading the way for a new class of networked systems characterized by their complex interconnections, diversity of components, and interactions with the physical world. The potential benefits of these networked systems from environmental, economic, and social perspectives are however limited by the ability to secure them. There are currently many efforts under way to address the underlying security issues by ways of developing secure communication protocols and procedures. The PIs take a different point of view here: instead of developing methods to secure existing systems, they develop methods to design systems that are inherently secure and embedded with breach detection mechanisms. The PIs thus expand the secure-by-design philosophy popular in software engineering to the design of networked dynamical systems. In terms of applications, the proposed research program will significantly push forward the frontiers of secure design of cyber-physical systems, furthering their use across many domains including critical infrastructures such as transportation networks, power generation and distribution networks, water and gas distribution networks. The PIs will also contribute to educational outreach by developing new interactive modules, focusing on security issues of CPS for high school students, and mentoring graduate and undergraduate students.

The proposed program is built upon three overarching principles: (i) control the information given to the agents, (ii) embed the agents with hidden security measures, and (iii) make the dynamics robust and resilient. More specifically, for the first principle, the PIs propose to establish a theoretical framework with novel design methodologies that can localize and encode both information and objectives for the agents. For the second principle, the PIs propose to embed the networked system with security measures that allow to detect easily tampering with large signals and its effect on the agents. These security measures can be certain functions that are designed to maintain invariant values over time and are locally computable by the agents. For the third principle, the PIs propose to leverage robust control theory for dealing with small attack injections, and moreover, introduce a novel controllability notion with provable guarantees on state attack recovery. In terms of intellectual merit, the proposed program will unify and exploit research in mathematics, namely integrating the study of symmetries in dynamical systems, completely integrable systems, and Hamiltonian dynamics, with engineering tools to develop methods to detect and foil attacks on a system. As such, it will establish new and deepen existing connections between mathematics, especially geometry and dynamical systems, and the study of cyber-physical systems.

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.

The complexity of modern neural nets, with their millions of parameters and unprecedented computational demands, has been a major hurdle for the conventional approaches which had been successfully applied in machine learning over the past decades. This project aims to develop new mathematical and computational foundations for the analysis and design of these systems through a radically new conceptualization of their architectures as continuous dynamical systems. The key pillar of this framework is the idealization of depth as a continuum of layers and width as a continuum of neurons. Infinitesimal abstractions of this type have successfully unlocked many disciplines throughout the twentieth century, including probability, optimization, control, and many more. This collaborative project involving UIUC and MIT will push the boundaries of the theory and practice of deep learning, while sparking sustained interactions between the communities of electrical engineering, mathematics, statistics, and theoretical computer science. The project will also have broad impacts through a deliberate approach to education and training. The education and outreach activities will include research opportunities for undergraduate students at both institutions, as well as an exchange program to foster the collaboration and exchange of ideas.

This project on Analysis and Geometry of Neural Dynamical Systems is developing the mathematical foundations of deep learning by synthesizing tools from probability, statistics, dynamical systems, geometric analysis, partial differential equations, and optimal transport. The research program is articulated around three major directions: (1) continuous models of neural dynamical systems; (2) discretization schemes; and (3) algorithms. The first direction is focusing on characterizing the tradeoffs between the expressive power and complexity of idealized infinitely wide and deep neural nets. The second direction builds on these continuous abstractions to develop, from first principles, mathematically rigorous and practically implementable techniques for analyzing large but finite neural nets. The third direction emphasizes algorithmic and computational aspects, such as the computational complexity of numerical methods, stability, and implicit regularization, using a novel synthesis of analytic and geometric methods developed as part of the project.

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.