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Mathematics > Optimization and Control

arXiv:2108.12070 (math)
[Submitted on 26 Aug 2021 (v1), last revised 30 Dec 2022 (this version, v2)]

Title:Mirror frameworks for relatively Lipschitz and monotone-like variational inequalities

Authors:Hui Zhang, Yu-Hong Dai
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Abstract:Nonconvex-nonconcave saddle-point optimization in machine learning has triggered lots of research for studying non-monotone variational inequalities (VI). In this work, we introduce two mirror frameworks, called mirror extragradient method and mirror extrapolation method, for approximating solutions to relatively Lipschitz and monotone-like VIs. The former covers the well-known Nemirovski's mirror prox method and Nesterov's dual extrapolation method, and the recently proposed Bregman extragradient method; all of them can be reformulated into a scheme that is very similar to the original form of extragradient method. The latter includes the operator extrapolation method and the Bregman extrapolation method as its special cases. The proposed mirror frameworks allow us to present a unified and improved convergence analysis for all these existing methods under relative Lipschitzness and monotone-like conditions that may be the currently weakest assumptions.
Comments: 26 pages, 2 figures
Subjects: Optimization and Control (math.OC)
Cite as: arXiv:2108.12070 [math.OC]
  (or arXiv:2108.12070v2 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.2108.12070

Submission history

From: Hui Zhang [view email]
[v1] Thu, 26 Aug 2021 23:58:39 UTC (24 KB)
[v2] Fri, 30 Dec 2022 02:57:00 UTC (24 KB)
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