Convergence of gradient descent-ascent analyzed as a Newtonian dynamical system with dissipation

Abstract

A dynamical system is defined in terms of the gradient of a payoff function. Dynamical variables are of two types, ascent and descent. The ascent variables move in the direction of the gradient, while the descent variables move in the opposite direction. Dynamical systems of this form or very similar forms have been studied in diverse fields such as game theory, optimization, neural networks, and population biology. Gradient descent-ascent is approximated as a Newtonian dynamical system that conserves total energy, defined as the sum of the kinetic energy and a potential energy that is proportional to the payoff function. The error of the approximation is a residual force that violates energy conservation. If the residual force is purely dissipative, then the energy serves as a Lyapunov function, and convergence of bounded trajectories to steady states is guaranteed. A previous convergence theorem due to Kose and Uzawa required the payoff function to be convex in the descent variables, and concave in the ascent variables. Here the assumption is relaxed, so that the payoff function need only be globally `less convex' or `more concave' in the ascent variables than in the descent variables. Such relative convexity conditions allow the existence of multiple steady states, unlike the convex-concave assumption. When combined with sufficient conditions that imply the existence of a minimax equilibrium, boundedness of trajectories is also assured.