Towards characterizing the optimization landscape of games, this paper analyzes the stability of gradient-based dynamics near fixed points of two-player continuous games. We introduce the quadratic numerical range as a method to characterize the spectrum of game dynamics and prove the robustness of equilibria to variations in learning rates. By decomposing the game Jacobian into symmetric and skew-symmetric components, we assess the contribution of a vector field's potential and rotational components to the stability of differential Nash equilibria. Our results show that in zero-sum games, all Nash are stable and robust; in potential games, all stable points are Nash. For general-sum games, we provide a sufficient condition for instability. We conclude with a numerical example in which learning with timescale separation results in faster convergence.
翻译:本文旨在描述游戏的最佳景观, 分析两玩者连续游戏固定点附近基于梯度的动态的稳定性。 我们引入了二次数字范围, 以此来描述游戏动态的范围, 并证明对学习率变化的平衡性。 通过将游戏Jacobian 分解为对称和扭曲对称成分, 我们评估矢量字段的潜力和旋转组件对差异Nash均衡稳定的贡献。 我们的结果表明, 在零和游戏中, 所有Nash都是稳定和稳健的; 在潜在游戏中, 所有稳定点都是 Nash 。 对于普通和游戏, 我们为不稳定提供了充分的条件。 我们以一个数字例子来结束我们学习时间尺度分离的结果, 更快的融合。