AI 中文总结
本文研究FLBR-MWU动力学，确定其对偶间隙的几何收敛速率，通过实验对比表明该方法性能与OGDA相当或更优。
AI 中文摘要
我们重新审视由Fasoulakis等人[2022年AISTATS]提出的多权重更新（MWU）变体，即前瞻最佳响应MWU（FLBR-MWU）。该动力学基于额外梯度方法，调整了中间步骤和实际更新步骤使用不同学习率。目前已证明该算法具有渐近最后迭代收敛性，但尚未明确收敛速率。我们针对Fasoulakis等人提出的开放问题，确定了对偶间隙的具体收敛速率，具体为几何收敛速率，形式为$O(c^t)$，其中$c<1$不随时间变化但依赖于博弈参数，如雅可比矩阵的最大特征值。我们还将理论分析与乐观梯度下降-上升（OGDA）的实验对比作为补充，OGDA是求解零和博弈的最优最后迭代方法之一，结果表明FLBR-MWU方法的性能与OGDA相当，部分情况下优于OGDA。
英文摘要
We revisit a variant of Multiplicative Weights Update (MWU), defined recently by Fasoulakis et al. [AISTATS; 2022], and denoted as Forward Looking Best Response MWU (FLBR-MWU). These dynamics are based on the approach of extra-gradient methods, with the tweak of using different learning rates in the intermediate step and the actual update step. So far, it has been proved that this algorithm attains asymptotic last-iterate convergence but no explicit rate has been known. We answer the open question from Fasoulakis et al. by establishing a concrete convergence rate for the duality gap. In particular, we show a geometric convergence rate, of the form $O(c^t)$, where $c<1$ is independent of time but dependent on game parameters, such as the maximum eigenvalue of the Jacobian matrix. We also complement our theoretical analysis with an experimental comparison to OGDA (Optimistic Gradient Descent-Ascent), which ranks among the best last-iterate methods for solving zero-sum games. We demonstrate that the performance of the FLBR-MWU method matches or, in some cases, outperforms OGDA.
Comments27 pages, 11 figures
