Title: STay-On-the-Ridge (STON'R): Guaranteed Convergence to Local Minimax Equilibrium in Nonconvex-Nonconcave Games. 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 [Flokas et al. '19, Hsieh et al. '21]. The only known convergence guarantees hold under the strong assumption that the initialization is very close to a local min-max equilibrium [Wang et al. '19]. 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. 