用微分方程解析Adam-DA在零和博弈中的动态，发现动量作用与优化问题相反。
Understanding Dynamics of Adam in Zero-Sum Games: An ODE Approach
- 通过连续时间微分方程逼近Adam-DA离散动力学
- 发现一阶、二阶动量参数在博弈中作用与优化时相反
- 实验验证该反向动量效应在多模型多数据集上均成立
Adam在训练神经网络中的显著成功，自然推动了其下降-上升变体Adam-DA在零和博弈中的广泛应用。尽管其在实践中广受欢迎，但对Adam-DA的严格理论理解仍滞后。本文推导出描述Adam-DA连续时间极限的常微分方程（ODE），这些方程能精确逼近离散时间下的动态行为，为分析其在零和博弈中的表现提供了可处理的解析框架。利用该方法，我们研究了Adam-DA的两个基本方面：局部收敛性与隐式梯度正则化。分析表明，在零和博弈中，一阶与二阶动量参数的作用恰好与最小化问题中的已知效果相反。通过在多种架构和数据集上的生成对抗网络（GAN）实验，验证了上述预测，展示了该反向动量效应的实际意义。
原文摘要 · Abstract (English)
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.
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