提出GOMA方法，实现单调变分不等式求解的加速稳定收敛。
Accelerated and Stable Convergence with Anchored Generalized Optimistic Method
- 引入锚定项与双时标优化更新，改进经典外梯度法。
- 确定性下达到最优加速率 $O(1/k^2)$，随机下达 $O(1/\\/sqrt{k})$。
- 适用于无方差缩减的在线/随机场景，适合优化算法研究者。
我们研究用于求解最小-最大优化中单调变分不等式的一阶方法。经典方法如外梯度法每轮需两次梯度查询，限制了其在在线和随机设置中的分析与应用。本文提出一类带锚定的广义乐观方法（GOMA），结合双时标乐观更新与受Halpern迭代启发的锚定项。在确定性设定下，GOMA对单调Lipschitz算子实现了关于平方梯度范数的最优加速最后迭代率 $O(1/k^2)$。在具有无界方差的随机设定中，简化版单次调用变体达到最后迭代收敛率 $O(1/\sqrt{k})$。据我们所知，这是首个在无约束设定下，无需方差缩减或增长批大小的随机单调Lipschitz变分不等式收敛保证。
原文摘要 · Abstract (English)
We study first-order methods for solving monotone variational inequalities arising in min-max optimization. Classical approaches such as the extragradient method rely on two gradient queries per iteration, which limits their analysis and applicability in the online and stochastic settings. We propose a family of Generalized Optimistic Methods with Anchoring (GOMA), which combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration. In the deterministic setting, GOMA achieves the optimal accelerated last-iterate rate $O(1/k^2)$ on the squared gradient norm for monotone Lipschitz operators. In the stochastic setting with unbounded variance, a simplified single-call variant of GOMA achieves a last-iterate convergence rate of $O(1/\sqrt{k})$ on the squared gradient norm. To the best of our knowledge, this is the first such guarantee for stochastic monotone Lipschitz variational inequalities in the unconstrained setting without variance reduction or growing batches.
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