通过一阶方法获得光滑非凸-非凹极小极大问题的博弈驻点
Obtaining Game-Stationary Points for Smooth Nonconvex-Nonconcave Minimax Problems via First-Order Methods
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- University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
- University of Minnesota–Twin Cities（明尼苏达大学双城分校）
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中文总结 AI 辅助
针对一般非凸-非凹极小极大问题，提出一种在局部逆Lipschitz正则性条件下仅需一阶迭代即可找到ε-驻点的方法，复杂度为Õ(ε⁻²)，无需全局最优性或PL/KL/Minty型条件，扩展了可处理问题类别。
中文摘要 AI 辅助
极小极大优化是机器学习、鲁棒优化和博弈论中的一个基本框架，然而在没有额外结构假设的情况下，寻找一般非凸-非凹极小极大问题的一阶驻点仍然具有挑战性。现有的保证通常依赖于全局PL型或KL型条件，这些条件将最大玩家驻点与全局内部最优性联系起来，或者依赖于Minty型条件，这些条件对相对于参考解的博弈梯度场施加全局关系；局部KL变体放宽了前一种要求，但通常需要在接近最优的区域内进行初始化和跟踪。这些条件在许多应用中可能难以满足。相比之下，我们开发了一种一阶方法，在近似最大玩家驻点周围的局部逆Lipschitz正则性条件以及惩罚子水平集的紧性条件下，该方法在$\widetilde{O}(\epsilon^{-2})$次一阶迭代内找到$\epsilon$-驻点。我们的条件对内部最大化问题的驻点不施加最优性要求：它们不需要是全局甚至局部最大化的。我们进一步提供了所需正则性的充分条件。在无约束设置中，它源于最大化变量Hessian在驻点附近的均匀非奇异性；对于带约束的上层Moreau包络，它源于标准的KKT正则性条件。这些结果为上述PL型、KL型或Minty型框架未覆盖的非凸-非凹极小极大问题类别建立了第一阶复杂度保证。
英文摘要
Minimax optimization is a fundamental framework in machine learning, robust optimization, and game theory, yet finding first-order stationary points of general nonconvex-nonconcave minimax problems remains challenging without additional structural assumptions. Existing guarantees often rely on global PL- or KL-type conditions that connect max-player stationarity to global inner optimality, or on Minty-type conditions that impose a global relation on the game gradient field relative to a reference solution; local KL variants relax the former requirement but typically require initialization and tracking within a near-optimal region. Such conditions may be difficult to satisfy in many applications. In contrast, we develop a first-order method that finds an $ε$-stationary point within $\widetilde{O}(ε^{-2})$ first-order iterations under a local inverse-Lipschitz regularity condition around approximate max-player stationary points, together with a compactness condition on a penalty sublevel set. Our condition places no optimality requirement on stationary points of the inner maximization problem: they need not be globally, or even locally, maximizing. We further provide sufficient conditions for the required regularity. In the unconstrained setting, it follows from uniform nonsingularity of the maximization-variable Hessian near stationary points; for constrained upper Moreau envelopes, it follows from standard KKT regularity conditions. These results establish first-order complexity guarantees for classes of nonconvex-nonconcave minimax problems not covered by the above PL-, KL-, or Minty-type frameworks.
