In a recent series of papers it has been established that variants of Gradient Descent/Ascent and Mirror Descent exhibit last iterate convergence in convex-concave zero-sum games. Specifically, \cite{DISZ17, LiangS18} show last iterate convergence of the so called "Optimistic Gradient Descent/Ascent" for the case of \textit{unconstrained} min-max optimization. Moreover, in \cite{Metal} the authors show that Mirror Descent with an extra gradient step displays last iterate convergence for convex-concave problems (both constrained and unconstrained), though their algorithm does not follow the online learning framework; it uses extra information rather than \textit{only} the history to compute the next iteration. In this work, we show that "Optimistic Multiplicative-Weights Update (OMWU)" which follows the no-regret online learning framework, exhibits last iterate convergence locally for convex-concave games, generalizing the results of \cite{DP19} where last iterate convergence of OMWU was shown only for the \textit{bilinear case}. We complement our results with experiments that indicate fast convergence of the method.

中文翻译：

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无悔学习中的最后迭代收敛：凸凹景观的约束最小-最大优化

在最近的一系列论文中，已经确定梯度下降/上升和镜像下降的变体在凸凹零和游戏中表现出最后迭代收敛。具体来说，\cite{DISZ17, LiangS18} 展示了 \textit{unconstrained} min-max 优化情况下所谓的“Optimistic Gradient Descent/Ascent”的最后一次迭代收敛。此外，在 \cite{Metal} 中，作者展示了具有额外梯度步骤的 Mirror Descent 显示了凸凹问题（约束和无约束）的最后迭代收敛，尽管他们的算法不遵循在线学习框架；它使用额外的信息而不是 \textit{only} 历史来计算下一次迭代。在这项工作中，我们展示了“乐观乘法权重更新（OMWU）” 它遵循无后悔在线学习框架，在局部展示了凸凹游戏的最后一次迭代收敛，概括了 \cite{DP19} 的结果，其中 OMWU 的最后一次迭代收敛仅针对 \textit{bilinear case} 显示。我们用表明该方法快速收敛的实验来补充我们的结果。