Motivated by applications in Game Theory, Optimization, and Generative Adversarial Networks, recent work of Daskalakis et al \cite{DISZ17} and follow-up work of Liang and Stokes \cite{LiangS18} have established that a variant of the widely used Gradient Descent/Ascent procedure, called "Optimistic Gradient Descent/Ascent (OGDA)", exhibits last-iterate convergence to saddle points in {\em unconstrained} convex-concave min-max optimization problems. We show that the same holds true in the more general problem of {\em constrained} min-max optimization under a variant of the no-regret Multiplicative-Weights-Update method called "Optimistic Multiplicative-Weights Update (OMWU)". This answers an open question of Syrgkanis et al \cite{SALS15}. The proof of our result requires fundamentally different techniques from those that exist in no-regret learning literature and the aforementioned papers. We show that OMWU monotonically improves the Kullback-Leibler divergence of the current iterate to the (appropriately normalized) min-max solution until it enters a neighborhood of the solution. Inside that neighborhood we show that OMWU becomes a contracting map converging to the exact solution. We believe that our techniques will be useful in the analysis of the last iterate of other learning algorithms.

中文翻译：

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最后迭代收敛：零和博弈和受约束的最小-最大优化

受博弈论、优化和生成对抗网络中的应用的启发，Daskalakis 等人 \cite{DISZ17} 的近期工作以及 Liang 和 Stokes \cite{LiangS18} 的后续工作已经确定，广泛使用的梯度下降的变体/Ascent 程序，称为“乐观梯度下降/上升（OGDA）”，在{\em unconstrained} 凸凹最小-最大优化问题中表现出对鞍点的最后迭代收敛。我们表明，在称为“乐观乘法权重更新（OMWU）”的无后悔乘法权重更新方法的变体下，更一般的{\em 约束}最小-最大优化问题也是如此。这回答了 Syrgkanis 等人的一个悬而未决的问题 \cite{SALS15}。我们结果的证明需要与无悔学习文献和上述论文中存在的技术完全不同的技术。我们表明 OMWU 单调地将当前迭代的 Kullback-Leibler 散度提高到（适当归一化的）最小-最大解决方案，直到它进入解决方案的邻域。在该邻域内，我们表明 OMWU 成为收敛到精确解的收缩图。我们相信我们的技术将有助于分析其他学习算法的最后一次迭代。在该邻域内，我们表明 OMWU 成为收敛到精确解的收缩图。我们相信我们的技术将有助于分析其他学习算法的最后一次迭代。在该邻域内，我们表明 OMWU 成为收敛到精确解的收缩图。我们相信我们的技术将有助于分析其他学习算法的最后一次迭代。