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We propose a study of structured non-convex non-concave min-max problemswhich goes beyond standard first-order approaches. Inspired by the tightunderstanding established in recent works [Adil et al., 2022, Lin and Jordan,2022b], we develop a suite of higher-order methods which show the improvementsattainable beyond the monotone and Minty condition settings. Specifically, weprovide a new understanding of the use of discrete-time $p^{th}$-order methodsfor operator norm minimization in the min-max setting, establishing an$O(1/\epsilon^\frac{2}{p})$ rate to achieve $\epsilon$-approximatestationarity, under the weakened Minty variational inequality condition ofDiakonikolas et al. [2021]. We further present a continuous-time analysisalongside rates which match those for the discrete-time setting, and ourempirical results highlight the practical benefits of our approach overfirst-order methods.

Beyond first-order methods for non-convex non-concave min-max optimization