CAREER: Foundations of Scalable, Fast, and Online Decentralized Manifold Optimization in Multi-Agent Networks

Manifold optimization is instrumental in control and engineering applications, ranging from trajectory optimization in robotics and safe reinforcement learning to subspace estimation, geometric deep learning, and adaptive fine-tuning of large language models (LLMs). Despite recent advances, most of the research on manifold optimization is limited to centralized approaches not implementable in multi-agent systems. This CAREER project takes a substantial step towards the development and adoption of decentralized manifold optimization (DMO) in large-scale, multi-agent optimization by overcoming three fundamental challenges: scalability, efficiency, and adaptation to dynamic environments. The PI proposes three integrated research thrusts to address these challenges. (i) Thrust 1 will fundamentally advance the scalability of DMO by innovating decentralized retraction-free methods that are computationally efficient, and it will also extend the theory of retraction-free methods to bilevel DMO. (ii) Thrust 2 will provide a systematic understanding of acceleration to improve the iteration complexity of DMO algorithms. This research will establish provably fast convergence guarantees for accelerated DMO and further expand this theory to the zero-order setting using smoothing techniques to generate high-fidelity gradient estimators. (iii) Thrust 3 will pioneer online DMO to contextualize multi-agent Riemannian optimization in dynamic, unpredictable environments. It will investigate projection-free methods to replace costly projection operations with efficient oracles and further extend the study to functional constrained online DMO by developing decentralized Riemannian augmented Lagrangian methods. The proposed research in this CAREER project will produce and draw upon novel scientific tools in Riemannian optimization, multi-agent systems and control, high-dimensional statistics and concentration inequalities. The research agenda also includes a concrete evaluation plan of the proposed algorithms on fine-tuning of LLMs and active mapping/planning with robot teams. The PI has an integrated education plan to engage high school and undergraduate students in research and organize STEM field trips for middle school students. 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.