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. Existing results are limited to potential games with stringent structural assumptions and entail exponential convergence times in $1/\epsilon$. Unaddressed so far, we tackle general potential games and prove the first finite-time convergence to an $\epsilon$-efficient NE. In particular, by using a problem-dependent analysis, our bound depends polynomially on $1/\epsilon$. Furthermore, we provide two extensions of our convergence result: 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：本文研究了在势博弈中，对数线性学习算法收敛至$\epsilon$-效率纳什均衡（NE）所需的时间。在此类博弈中，效率纳什均衡被定义为势函数的最大化点。现有研究结果局限于具有严格结构假设的势博弈，且收敛时间在$1/\epsilon$上呈指数级增长。针对此前尚未解决的一般性势博弈问题，我们首次证明了算法可在有限时间内收敛至$\epsilon$-效率纳什均衡。特别地，通过采用问题依赖性分析，我们得出的收敛时间上界对$1/\epsilon$具有多项式依赖性。此外，我们提供了收敛结果的两种扩展：首先，我们证明了一种每轮所需效用反馈减少$A$倍的对数线性学习变体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（如学习规则调整或效用噪声干扰），其收敛保证仍具有鲁棒性。