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Min-max optimization problems involving nonconvex-nonconcave objectives havefound important applications in adversarial training and other multi-agentlearning settings. Yet, no known gradient descent-based method is guaranteed toconverge to (even local notions of) min-max equilibrium in thenonconvex-nonconcave setting. For all known methods, there exist relativelysimple objectives for which they cycle or exhibit other undesirable behaviordifferent from converging to a point, let alone to some game-theoreticallymeaningful one~\cite{flokas2019poincare,hsieh2021limits}. The only knownconvergence guarantees hold under the strong assumption that the initializationis very close to a local min-max equilibrium~\cite{wang2019solving}. Moreover,the afore-described challenges are not just theoretical curiosities. All knownmethods are unstable in practice, even in simple settings.  We propose the first method that is guaranteed to converge to a local min-maxequilibrium for smooth nonconvex-nonconcave objectives. Our method issecond-order and provably escapes limit cycles as long as it is initialized atan easy-to-find initial point. Both the definition of our method and itsconvergence analysis are motivated by the topological nature of the problem. Inparticular, 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 orthe projected gradient of the objective, but is designed to satisfy atopological property that guarantees the avoidance of cycles and implies itsconvergence.

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