We propose a novel variant of the \emph{multiplicative weights update method} with forward-looking best-response strategies, that guarantees last-iterate convergence for \emph{zero-sum games} with a unique \emph{Nash equilibrium}. Particularly, we show that the proposed algorithm converges to an $\eta^{1/\rho}$-approximate Nash equilibrium, with $\rho > 1$, by decreasing the Kullback-Leibler divergence of each iterate by a rate of at least $\Omega(\eta^{1+\frac{1}{\rho}})$, for sufficiently small learning rate $\eta$. When our method enters a sufficiently small neighborhood of the solution, it becomes a contraction and converges to the Nash equilibrium of the game. Furthermore, we perform an experimental comparison with the recently proposed optimistic variant of the multiplicative weights update method, by \cite{Daskalakis2019LastIterateCZ}, which has also been proved to attain last-iterate convergence. Our findings reveal that our algorithm offers substantial gains both in terms of the convergence rate and the region of contraction relative to the previous approach.
翻译:我们提出了一个具有前瞻性最佳应对战略的新变式,即 \ emph{ 倍增加权更新法, 保证 \ emph{ 零和游戏} 与 { 纳什 平衡} 的独特 。 特别是, 我们显示, 提议的算法与 $ { 1/\\ rh} 相近的纳什平衡相融合, 以 $ > 1 美元为单位, 通过降低 Kullback- Leiber 差异, 以至少 $ / Omega (\ ⁇ 1 { { { { { { { { { { 1 { { { { ⁇ } { { { { ⁇ { { ⁇ { ⁇ } 。 } 保证 足够小的学习率, $ 。 当我们的方法进入一个足够小的解决方案附近时,, 它就会变成收缩, 与 游戏的纳什 平衡一致 。 此外, 我们用最近提出的多复制权重的比较变式方法进行了实验性比较比较比较比较比较,, 通过\, 通过\ 。