International Journal of Mathematics and Systems Science 2024, 7(10); https://doi.org/10.18686/ijmss.v7i10.10139

This study introduces two first-order algorithms for high-dimensional minimax optimization: Accelerated Momentum Descent Ascent (AMDA) and Accelerated Variance-Reduced Gradient Descent Ascent (AVRGDA). These methods aim to address common challenges in nonconvex optimization, such as slow convergence and computational complexity. AMDA leverages momentum-driven techniques to smooth the optimization path, reducing oscillations and improving convergence speed, particularly in nonconvex-strongly-concave problems. AVRGDA incorporates adaptive learning rates that dynamically adjust according to gradient norms, enhancing the efficiency of variance reduction and handling complex optimization tasks in high-dimensional spaces. Through experiments in adversarial training and large-scale logistic regression, these methods demonstrate superior performance in terms of training time, robustness, and computational cost compared to traditional first-order methods. Theoretical analysis shows that AMDA and AVRGDA achieve convergence rates of O(ϵ−3)and O(ϵ−2.5) respectively in high-dimensional, nonconvex minimax problems, confirming their efficiency and robustness in practical applications.