Convergence of Learning Dynamics in Stackelberg Games

TL;DR

This paper analyzes the convergence properties of learning dynamics in Stackelberg games, establishing conditions for equilibrium convergence and proposing algorithms with proven convergence guarantees, including applications to training GANs.

Contribution

It introduces new gradient-based algorithms for Stackelberg games with convergence guarantees and applies these methods to improve training of generative adversarial networks.

Findings

Stable critical points correspond to Stackelberg equilibria in zero-sum games.

Proposed algorithms converge to Stackelberg equilibria under certain conditions.

Numerical experiments validate theoretical results and improve GAN training.

Abstract

This paper investigates the convergence of learning dynamics in Stackelberg games. In the class of games we consider, there is a hierarchical game being played between a leader and a follower with continuous action spaces. We establish a number of connections between the Nash and Stackelberg equilibrium concepts and characterize conditions under which attracting critical points of simultaneous gradient descent are Stackelberg equilibria in zero-sum games. Moreover, we show that the only stable critical points of the Stackelberg gradient dynamics are Stackelberg equilibria in zero-sum games. Using this insight, we develop a gradient-based update for the leader while the follower employs a best response strategy for which each stable critical point is guaranteed to be a Stackelberg equilibrium in zero-sum games. As a result, the learning rule provably converges to a Stackelberg equilibria given an initialization in the region of attraction of a stable critical point. We then consider a follower employing a gradient-play update rule instead of a best response strategy and propose a two-timescale algorithm with similar asymptotic convergence guarantees. For this algorithm, we also provide finite-time high probability bounds for local convergence to a neighborhood of a stable Stackelberg equilibrium in general-sum games. Finally, we present extensive numerical results that validate our theory, provide insights into the optimization landscape of generative adversarial networks, and demonstrate that the learning dynamics we propose can effectively train generative adversarial networks.