On the Linear Convergence of Extragradient Methods for Nonconvex–Nonconcave Minimax Problems

Saeed Hajizadeh, Haihao Lu, Benjamin Grimmer
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引用次数: 5

Abstract

Recently, minimax optimization has received renewed focus due to modern applications in machine learning, robust optimization, and reinforcement learning. The scale of these applications naturally leads to the use of first-order methods. However, the nonconvexities and nonconcavities present in these problems, prevents the application of typical gradient descent/ascent, which is known to diverge even in bilinear problems. Recently, it was shown that the proximal point method (PPM) converges linearly for a family of nonconvex–nonconcave problems. In this paper, we study the convergence of a damped version of the extragradient method (EGM), which avoids potentially costly proximal computations, relying only on gradient evaluation. We show that the EGM converges linearly for smooth minimax optimization problems satisfying the same nonconvex–nonconcave condition needed by the PPM. Funding: H. Lu was supported by The University of Chicago Booth School of Business Benjamin Grimmer was supported by Johns Hopkins Applied Mathematics and Statistics Department.
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非凸非凹极小极大问题的外聚方法的线性收敛性
最近,极大极小优化由于在机器学习、鲁棒优化和强化学习中的现代应用而重新受到关注。这些应用的规模自然导致使用一阶方法。然而,这些问题中存在的非凸性和非凹性阻碍了典型的梯度下降/上升方法的应用,这种方法即使在双线性问题中也是发散的。最近,证明了近点法对于一类非凸非凹问题是线性收敛的。在本文中,我们研究了一种阻尼版本的extragradient方法(EGM)的收敛性,该方法避免了潜在的昂贵的近端计算,仅依赖于梯度评估。我们证明了EGM对于满足PPM所需的相同非凸非凹条件的光滑极小极大优化问题是线性收敛的。资助:H. Lu由芝加哥大学布斯商学院资助;Benjamin Grimmer由约翰霍普金斯大学应用数学与统计学系资助。
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