Min-max optimization problems (i.e., min-max games) have been attracting a great deal of attention because of their applicability to a wide range of machine learning problems. Although significant progress has been made recently, the literature to date has focused on games with independent strategy sets; little is known about solving games with dependent strategy sets, which can be characterized as min-max Stackelberg games. We introduce two first-order methods that solve a large class of convex-concave min-max Stackelberg games, and show that our methods converge in polynomial time. Min-max Stackelberg games were first studied by Wald, under the posthumous name of Wald's maximin model, a variant of which is the main paradigm used in robust optimization, which means that our methods can likewise solve many convex robust optimization problems. We observe that the computation of competitive equilibria in Fisher markets also comprises a min-max Stackelberg game. Further, we demonstrate the efficacy and efficiency of our algorithms in practice by computing competitive equilibria in Fisher markets with varying utility structures. Our experiments suggest potential ways to extend our theoretical results, by demonstrating how different smoothness properties can affect the convergence rate of our algorithms.

翻译：极小极大优化问题（即极小极大博弈）因其在广泛机器学习问题中的适用性而备受关注。尽管近年来取得了显著进展，但现有文献主要关注具有独立策略集的博弈；对于具有依赖策略集的博弈（可表征为极小极大Stackelberg博弈）的求解方法知之甚少。我们提出了两种一阶方法，用于求解一大类凸-凹极小极大Stackelberg博弈，并证明我们的方法能在多项式时间内收敛。极小极大Stackelberg博弈最早由Wald研究，其遗作中提出的Wald极大极小模型是鲁棒优化中使用的主要范式变体之一，这意味着我们的方法同样能求解许多凸鲁棒优化问题。我们观察到Fisher市场中的竞争均衡计算也构成一个极小极大Stackelberg博弈。此外，通过计算具有不同效用结构的Fisher市场中的竞争均衡，我们证明了算法在实际中的有效性和高效性。实验通过展示不同光滑性特性如何影响算法收敛速度，为我们理论结果的拓展提供了潜在方向。