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.

翻译：我们提出了一项超越标准一阶方法的、针对结构化非凸非凹极小极大问题的研究。受近期工作[Adil等人，2022；Lin和Jordan，2022b]中建立深刻理解的启发，我们开发了一套高阶方法，展示了在单调和Minty条件设置之外可实现的改进。具体来说，我们提供了在极小极大设置中使用离散时间$p$阶方法进行算子范数最小化的新理解，在Diakonikolas等人[2021]的弱化Minty变分不等式条件下，建立了达到$\epsilon$近似稳定性的$O(1/\epsilon^\frac{2}{p})$速率。我们还给出了与离散时间设置速率相匹配的连续时间分析，并且实验结果表明我们的方法相比一阶方法具有实际优势。