This paper investigates the convergence time of log-linear learning to an $ε$-efficient Nash equilibrium in potential games, where an efficient Nash equilibrium is defined as the maximizer of the potential function. Previous literature provides asymptotic convergence rates to efficient Nash equilibria, and existing finite-time rates are limited to potential games with further assumptions such as the interchangeability of players. We prove the first finite-time convergence to an $ε$-efficient Nash equilibrium in general potential games. Our bounds depend polynomially on $1/ε$, an improvement over previous bounds for subclasses of potential games that are exponential in $1/ε$. We then strengthen our convergence result in two directions: first, we show that a variant of log-linear learning requiring a constant factor less feedback on the utility per round enjoys a similar convergence time; second, we demonstrate the robustness of our convergence guarantee if log-linear learning is subject to small perturbations such as alterations in the learning rule or noise-corrupted utilities.

翻译：本文研究了在势博弈中，对数线性学习收敛至 $ε$-有效纳什均衡的时间，其中有效纳什均衡被定义为势函数的最大化点。现有文献提供了收敛至有效纳什均衡的渐近收敛速率，而现有的有限时间收敛结果仅限于具有额外假设（如玩家的可交换性）的势博弈。我们首次证明了在一般势博弈中有限时间内收敛至 $ε$-有效纳什均衡。我们的收敛界对 $1/ε$ 具有多项式依赖性，这改进了先前针对势博弈子类所给出的、对 $1/ε$ 呈指数依赖性的收敛界。随后，我们从两个方向强化了收敛结果：首先，我们证明了一种变体的对数线性学习（其在每轮中对效用所需的反馈信息减少一个常数因子）具有相似的收敛时间；其次，我们证明了即使对数线性学习受到微小扰动（例如学习规则的改变或效用被噪声干扰），我们的收敛保证依然具有鲁棒性。