Last iterate convergence in no-regret learning: constrained min-max optimization for convex-concave landscapes
Last iterate convergence in no-regret learning: constrained min-max optimization for convex-concave landscapes
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发表时间：
2020-02
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通讯作者：
Qi Lei;Sai Ganesh Nagarajan;Ioannis Panageas;Xiao Wang
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作者：
Qi Lei;Sai Ganesh Nagarajan;Ioannis Panageas;Xiao Wang
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凹凸零和游戏中表现出最后的收敛性。具体来说，\cite{DISZ 17，Liang S 18}显示了最后的收敛的所谓的“乐观梯度下降/上升”的情况下\textit{无约束}最小-最大优化。此外，在\cite{Metal}中，作者证明了具有额外梯度步长的镜像下降显示了凹凸问题（约束和无约束）的最后一次收敛，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\texit {only}历史来计算下一次迭代。在这项工作中，我们表明，“乐观的多重权重更新（OMWU）”，遵循无遗憾的在线学习框架，表现出最后的局部收敛的凹凸游戏，推广的结果\cite{DP 19}，其中OMWU的最后的收敛只显示为\textit{双线性的情况}。我们补充我们的结果与实验表明快速收敛的方法。
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.