Collaborative Research: CIF: Medium: Post-Modern Min-Max Optimization Theory: Departure from Classical Minimization Theory

Min-max optimization underpins technologies ranging from generative Artificial Intelligence (AI) to large-scale reinforcement learning, yet today’s methods remain slow and unreliable for many real-world tasks. This suboptimality stems from the traditional approach of adapting minimization techniques to the min-max setup, which necessarily overlooks the unique complexities inherent in min-max problems. This project fundamentally revises this approach, developing specialized theoretical frameworks and efficient algorithms tailored explicitly to min-max optimization. By establishing a deeper understanding of these unique characteristics, the proposed research will significantly enhance the efficiency and robustness of min-max optimization, directly impacting practical applications in machine learning and artificial intelligence. Technically, this project will first explore core theoretical foundations under idealized convex-concave conditions, emphasizing accelerated convergence through anchor-type algorithms and enhanced stochastic methods with relaxed assumptions. Building upon this, the project will also develop practical algorithms that are robust to realistic, non-ideal conditions, including methods for nonconvex problems, efficient sampling strategies for stochastic settings, and adaptive update rules. Additionally, the research will investigate efficient alternating-update strategies, proximal gradient-type methods, and applications to training deep neural networks. These efforts are anticipated to greatly enrich the mathematical tools of min-max optimization and to lead to the discovery of more practically efficient algorithms. 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.