STay-ON-the-Ridge: Guaranteed Convergence to Local Minimax Equilibrium in Nonconvex-Nonconcave Games

TL;DR

This paper introduces STay-ON-the-Ridge, a second-order method that guarantees convergence to local min-max equilibria in nonconvex-nonconcave games, overcoming previous instability and cycling issues.

Contribution

It presents the first guaranteed convergence method for smooth nonconvex-nonconcave objectives, leveraging topological properties rather than potential functions.

Findings

Guarantees convergence to local min-max equilibrium

Escapes limit cycles with proper initialization

Outperforms existing methods in stability and convergence

Abstract

Min-max optimization problems involving nonconvex-nonconcave objectives have found important applications in adversarial training and other multi-agent learning settings. Yet, no known gradient descent-based method is guaranteed to converge to (even local notions of) min-max equilibrium in the nonconvex-nonconcave setting. For all known methods, there exist relatively simple objectives for which they cycle or exhibit other undesirable behavior different from converging to a point, let alone to some game-theoretically meaningful one~\cite{flokas2019poincare,hsieh2021limits}. The only known convergence guarantees hold under the strong assumption that the initialization is very close to a local min-max equilibrium~\cite{wang2019solving}. Moreover, the afore-described challenges are not just theoretical curiosities. All known methods are unstable in practice, even in simple settings. We propose the first method that is guaranteed to converge to a local min-max equilibrium for smooth nonconvex-nonconcave objectives. Our method is second-order and provably escapes limit cycles as long as it is initialized at an easy-to-find initial point. Both the definition of our method and its convergence analysis are motivated by the topological nature of the problem. In particular, our method is not designed to decrease some potential function, such as the distance of its iterate from the set of local min-max equilibria or the projected gradient of the objective, but is designed to satisfy a topological property that guarantees the avoidance of cycles and implies its convergence.