Understanding Dynamics of Adam in Zero-Sum Games: An ODE Approach

The remarkable success of the Adam in training neural networks has naturally led to the widespread use of its descent-ascent counterpart, Adam-DA, for solving zero-sum games. Despite its popularity in practice, a rigorous theoretical understanding of Adam-DA still lags behind. In this paper, we derive ordinary differential equations (ODEs) that serve as continuous-time limits of the Adam-DA. These ODEs closely approximate the discrete-time dynamics of Adam-DA, providing a tractable analytical framework for understanding its behavior in zero-sum games. Using this ODE approach, we investigate two fundamental aspects of Adam-DA: local convergence and implicit gradient regularization. Our analysis reveals that the roles of the first- and second-order momentum parameters in zero-sum games are exactly the opposite of their well-documented effects in minimization problems. We validate these predictions through GAN experiments across multiple architectures and datasets, demonstrating the practical implications of this reversed momentum effect.

Adam is widely used to train neural networks. Its descent-ascent version, Adam-DA, is also commonly used in competitive learning problems such as GANs, where two models learn against each other. However, we still lack a clear theory explaining how Adam-DA behaves in such games. This paper studies Adam-DA by deriving smooth continuous-time models that closely approximate its update process. These models make it easier to analyze the local convergence and implicit regularization of Adam-DA. Our main finding is surprising: the two momentum parameters in Adam-DA play roles that are exactly opposite to their known roles in ordinary minimization problems. Experiments on GANs with different architectures and datasets support this prediction. These results provide both theoretical insight and practical guidance for tuning Adam-DA.