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. T