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