Differential games, in particular two-player sequential zero-sum games (a.k.a. minimax optimization), have been an important modeling tool in applied science and received renewed interest in machine learning due to many recent applications, such as adversarial training, generative models and reinforcement learning. However, existing theory mostly focuses on convex-concave functions with few exceptions. In this work, we propose two novel Newton-type algorithms for nonconvex-nonconcave minimax optimization. We prove their local convergence at strict local minimax points, which are surrogates of global solutions. We argue that our Newton-type algorithms nicely complement existing ones in that (a) they converge faster to strict local minimax points; (b) they are much more effective when the problem is ill-conditioned; (c) their computational complexity remains similar. We verify the effectiveness of our Newton-type algorithms through experiments on training GANs which are intrinsically nonconvex and ill-conditioned. Our code is available at https://github.com/watml/min-max-2nd-order.

翻译：微分博弈，特别是两人顺序零和博弈（又称极小极大优化），已成为应用科学中重要的建模工具，并因对抗训练、生成模型和强化学习等近期应用而在机器学习领域重获关注。然而，现有理论主要聚焦于凸-凹函数，鲜有例外。本文针对非凸-非凹极小极大优化问题，提出两种新型牛顿型算法。我们证明了这些算法在严格局部极小极大点（全局解的替代指标）处的局部收敛性。研究表明，我们的牛顿型算法对现有方法形成良好补充，具体体现在：（a）在严格局部极小极大点处收敛更快；（b）在病态问题中效率显著提升；（c）计算复杂度仍保持相似水平。通过在本质上非凸且病态的生成对抗网络训练实验，我们验证了所提牛顿型算法的有效性。代码见https://github.com/watml/min-max-2nd-order。