This paper investigates the convergence time of log-linear learning to an $\epsilon$-efficient Nash equilibrium (NE) in potential games. In such games, an efficient NE 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. In this paper, we prove the first finite-time convergence to an $\epsilon$-efficient NE in general potential games. Our bounds depend polynomially on $1/\epsilon$, an improvement over previous bounds that are exponential in $1/\epsilon$ and only hold for subclasses of potential games. We then strengthen our convergence result in two directions: first, we show that a variant of log-linear learning that requires a factor $A$ 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：本文研究了势博弈中log-linear学习算法收敛至ε-高效纳什均衡的时间复杂度。在此类博弈中，高效纳什均衡被定义为势函数的最大化点。现有文献仅给出了收敛至高效纳什均衡的渐近速率，而既有的有限时间收敛结果仅限于满足玩家可互换性等附加假设的势博弈。本文首次证明了在一般势博弈中实现ε-高效纳什均衡的有限时间收敛性。所得收敛界对1/ε具有多项式依赖性，相较于先前仅适用于势博弈子类且对1/ε呈指数依赖的收敛界有显著改进。随后我们从两个方向强化了收敛结果：首先证明了一种每轮仅需1/A效用反馈的log-linear学习变体具有相似的收敛时间；其次论证了当学习规则存在微小扰动或效用函数受噪声干扰时，收敛保证仍具有鲁棒性。