全谱对策的基于一致性的方法的严密统一分析
A Tight and Unified Analysis of Gradient-Based Methods for a Whole Spectrum of Games
摘要：
我们考虑的可微游戏的目标是找到一个纳什均衡。机器学习社区最近开始使用梯度方法（GD）的变体。最好的例子是超梯度法（EG）、乐观梯度法（OG）和共识优化（CO），它们在双线性博弈等情况下具有线性收敛性，而标准的GD是失败的。这些相对较新的方法的全部好处是不知道的，因为没有统一的分析强单调和双线性游戏。我们提供了新的分析EG的局部和全局收敛性和使用是为了得到一个更严格的全局收敛速度OG和CO。我们的分析涵盖了整个范围的设置之间的双线性和强单调游戏。它揭示了这些方法在这些极端情况下通过不同的机制收敛;在两者之间，它利用了给定问题的最有利机制。然后，我们证明了EG实现了最佳的速率为广泛的一类算法与任何数量的外推。我们严格的分析EG的收敛速度在游戏中表明，不像凸最小化，EG可能会比GD快得多。
英文摘要：
We consider differentiable games where the goal is to find a Nash equilibrium. The machine learning community has recently started using variants of the gradient method (GD). Prime examples are extragradient (EG), the optimistic gradient method (OG) and consensus optimization (CO), which enjoy linear convergence in cases like bilinear games, where the standard GD fails. The full benefits of theses relatively new methods are not known as there is no unified analysis for both strongly monotone and bilinear games. We provide new analyses of the EG's local and global convergence properties and use is to get a tighter global convergence rate for OG and CO. Our analysis covers the whole range of settings between bilinear and strongly monotone games. It reveals that these methods converge via different mechanisms at these extremes; in between, it exploits the most favorable mechanism for the given problem. We then prove that EG achieves the optimal rate for a wide class of algorithms with any number of extrapolations. Our tight analysis of EG's convergence rate in games shows that, unlike in convex minimization, EG may be much faster than GD.
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