Our work revolves around Fictitious Play, one of the first iterative methods that is known to converge to a Nash equilibrium in zero-sum games. In recent years, there has been a revived interest, due to applications in various machine learning problems, which has motivated a line of work on its convergence properties and on proposing new variants of the initial algorithm. Our paper is along this direction and introduces one new variant, which we refer to as Almost Greedy Fictitious Play. The proposed algorithm greedily attempts to find the optimal stepsize at each iteration but its search space is constrained and includes almost all the line between the cumulative mixed strategy and the current best response. Our main result is that the method achieves an instance dependent convergence rate of $\mathcal{O}(1/T)$ with respect to the duality gap. This matches the rate of Continuous Fictitious Play, and offers an alternative to discretization. We complement our theoretical findings with experiments that demonstrate the effectiveness of the method.

翻译：我们的工作围绕虚构博弈展开，这是已知在零和博弈中收敛到纳什均衡的首批迭代方法之一。近年来，由于在各类机器学习问题中的应用，该领域重新引起了研究兴趣，并推动了关于其收敛特性以及提出初始算法新变种的一系列工作。我们的论文沿此方向，引入了一种新变体，我们称之为“近乎贪婪的虚构博弈”。所提出的算法在每个迭代中贪婪地尝试寻找最优步长，但其搜索空间受到约束，涵盖了累积混合策略与当前最优反应之间几乎所有的线段。我们的主要结果是，该方法在对偶间隙方面实现了依赖于实例的$\mathcal{O}(1/T)$收敛速率。这与连续虚构博弈的速率相匹配，并为离散化提供了一种替代方案。我们通过实验补充了理论发现，证明了该方法的有效性。