We study decentralized learning in two-player zero-sum discounted Markov games where the goal is to design a policy optimization algorithm for either agent satisfying two properties. First, the player does not need to know the policy of the opponent to update its policy. Second, when both players adopt the algorithm, their joint policy converges to a Nash equilibrium of the game. To this end, we construct a meta algorithm, dubbed as $\texttt{Homotopy-PO}$, which provably finds a Nash equilibrium at a global linear rate. In particular, $\texttt{Homotopy-PO}$ interweaves two base algorithms $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ via homotopy continuation. $\texttt{Local-Fast}$ is an algorithm that enjoys local linear convergence while $\texttt{Global-Slow}$ is an algorithm that converges globally but at a slower sublinear rate. By switching between these two base algorithms, $\texttt{Global-Slow}$ essentially serves as a ``guide'' which identifies a benign neighborhood where $\texttt{Local-Fast}$ enjoys fast convergence. However, since the exact size of such a neighborhood is unknown, we apply a doubling trick to switch between these two base algorithms. The switching scheme is delicately designed so that the aggregated performance of the algorithm is driven by $\texttt{Local-Fast}$. Furthermore, we prove that $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ can both be instantiated by variants of optimistic gradient descent/ascent (OGDA) method, which is of independent interest.

研究了两个玩家的零和博弈下去中心化学习问题，提出了一种名为Homotopy-PO的元算法，通过交替使用本地快速算法和全局缓慢算法，使得机器人之间的政策收敛到了较稳健的纳什均衡状态，同时深入分析了算法的收敛性和性能。

在马尔科夫博弈中我们能以线性速率找到纳什均衡吗？