Min-max optimization problems (i.e., min-max games) have attracted a great deal of attention recently as their applicability to a wide range of machine learning problems has become evident. In this paper, we study min-max games with dependent strategy sets, where the strategy of the first player constrains the behavior of the second. Such games are best understood as sequential, i.e., Stackelberg, games, for which the relevant solution concept is Stackelberg equilibrium, a generalization of Nash. One of the most popular algorithms for solving min-max games is gradient descent ascent (GDA). We present a straightforward generalization of GDA to min-max Stackelberg games with dependent strategy sets, but show that it may not converge to a Stackelberg equilibrium. We then introduce two variants of GDA, which assume access to a solution oracle for the optimal Karush Kuhn Tucker (KKT) multipliers of the games' constraints. We show that such an oracle exists for a large class of convex-concave min-max Stackelberg games, and provide proof that our GDA variants with such an oracle converge in $O(\frac{1}{\varepsilon^2})$ iterations to an $\varepsilon$-Stackelberg equilibrium, improving on the most efficient algorithms currently known which converge in $O(\frac{1}{\varepsilon^3})$ iterations. We then show that solving Fisher markets, a canonical example of a min-max Stackelberg game, using our novel algorithm, corresponds to buyers and sellers using myopic best-response dynamics in a repeated market, allowing us to prove the convergence of these dynamics in $O(\frac{1}{\varepsilon^2})$ iterations in Fisher markets. We close by describing experiments on Fisher markets which suggest potential ways to extend our theoretical results, by demonstrating how different properties of the objective function can affect the convergence and convergence rate of our algorithms.

翻译：最小- 最大优化问题( 即, 最小- 最大游戏 ) 最近引起人们的极大关注, 因为它们对一系列广泛的机器学习问题的可应用性已经变得很明显。 在本文中, 我们用依赖性战略组研究最小- 马克游戏, 第一个玩家的策略约束第二个玩家的行为。 这些游戏最好被理解为顺序游戏, 即 Stackelberg, 相关解决方案概念是Stackelberg 平衡, 纳什的概括化。 解决最小- 马克游戏的最受欢迎的算法之一是渐变( GDA) 。 GDA 向具有依赖性战略组的最小- 斯塔克堡游戏进行直观化的游戏。 但是, 我们引入两种GDA变式, 假设最优的Karush Kuhn Tark( KKT) 乘以游戏限制的乘数。 我们展示了这样一个变数, 用于一个大型的直流- 直角- 平方- 平方- 平方- 平方- 平方- 平方- 平方- 平方- 平方- 平方- 平方- 游戏, 显示我们目前已知的变现的变现的变现, 数字- 数字- 变现一个变现的变现的变现的变现, 变现的变的变现, 变的变数- 基- 基- 以 美元- 基- 基- 基- 基- 基- 变数- 基- 基- 基- 基- 基- 基- 基- 基- 变数 变数- 变的 变数- 变数- 变数- 变数- 变数 变数- 变数- 变数 变数 变数 变数 变 变 变 变 变 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数 变数