隐式中点梯度下降法：零和博弈的快速且无学习率收敛
Implicit Midpoint Gradient Descent: Fast and Learning rate free convergence for Zero-Sum Games
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中文总结 AI 辅助
研究无约束双线性零和博弈，引入由辛积分方法推导的隐式中点梯度下降规则，该方法继承连续时间动力学的强大特性，在零和博弈中同时具备有界轨道等特性，且计算实验显示其显著优于标准方法。
中文摘要 AI 辅助
我们研究无约束双线性零和博弈，这是在线学习、对抗优化和多智能体决策中的基础模型。我们引入隐式中点梯度下降规则，它通过辛积分方法从连续时间跟随正则化领导者动力学推导得出。我们证明该方法继承了连续时间动力学的一些强大特性，包括有界轨道、快速遍历收敛到纳什均衡以及与学习率无关的稳定性保证。这是无约束双线性零和博弈中首个同时具备这些特性的传统在线优化方法。最后，计算实验表明该方法显著优于标准方法，如乐观梯度下降和交替梯度下降。
英文摘要
We study unconstrained bilinear zero-sum games, a fundamental model in online learning, adversarial optimization, and multi-agent decision-making. We introduce the implicit midpoint gradient descent rule, which we derive from continuous-time follow-the-regularized leader dynamics via symplectic integration methods. We prove that implicit midpoint gradient descent inherits several powerful properties from the continuous-time dynamics, including bounded orbits, fast ergodic convergence to Nash equilibria, and learning-rate-independent stability guarantees. This is the first traditional online optimization approach to simultaneously achieve these properties in unconstrained bilinear zero-sum games. Finally, computational experiments demonstrate that the proposed method significantly outperforms the standard methods, optimistic and alternating gradient descent.
