AdaGDA: Faster Adaptive Gradient Descent Ascent Methods for Minimax Optimization

Abstract

In the paper, we propose a class of faster adaptive Gradient Descent Ascent (GDA) methods for solving the nonconvex-strongly-concave minimax problems by using the unified adaptive matrices, which include almost all existing coordinate-wise and global adaptive learning rates. In particular, we provide an effective convergence analysis framework for our adaptive GDA methods. Specifically, we propose a fast Adaptive Gradient Descent Ascent (AdaGDA) method based on the basic momentum technique, which reaches a lower gradient complexity of $\tilde{O}(\kappa^4\epsilon^{-4})$ for finding an $\epsilon$-stationary point without large batches, which improves the existing results of the adaptive GDA methods by a factor of $O(\sqrt{\kappa})$. Moreover, we propose an accelerated version of AdaGDA (VR-AdaGDA) method based on the momentum-based variance reduced technique, which achieves a lower gradient complexity of $\tilde{O}(\kappa^{4.5}\epsilon^{-3})$ for finding an $\epsilon$-stationary point without large batches, which improves the existing results of the adaptive GDA methods by a factor of $O(\epsilon^{-1})$. Moreover, we prove that our VR-AdaGDA method can reach the best known gradient complexity of $\tilde{O}(\kappa^{3}\epsilon^{-3})$ with the mini-batch size $O(\kappa^3)$. The experiments on policy evaluation and fair classifier learning tasks are conducted to verify the efficiency of our new algorithms.