Stability of Gradient Learning Dynamics in Continuous Games: Vector Action Spaces

2020 
Towards characterizing the optimization landscape of games, this paper analyzes the stability and spectrum of gradient-based dynamics near fixed points of two-player continuous games. We introduce the quadratic numerical range as a method to bound the spectrum of game dynamics linearized about local equilibria. We also analyze the stability of differential Nash equilibria and their robustness to variation in agent's learning rates. Our results show that by decomposing the game Jacobian into symmetric and anti-symmetric components, we can assess the contribution of vector field's potential and rotational components to the stability of the equilibrium. In zero-sum games, all differential Nash equilibria are stable; in potential games, all stable points are Nash. Furthermore, zero-sum Nash equilibria are robust in the sense that they are stable for all learning rates. For continuous games with general costs, we provide a sufficient condition for instability. We conclude with a numerical example that investigates how players with different learning rates can take advantage of rotational components of the game to converge faster.
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