Due to their importance in various emerging applications, efficient algorithms for solving minimax problems have recently received increasing attention. However, many existing algorithms require prior knowledge of the problem parameters in order to achieve optimal iteration complexity. In this paper, three completely parameter-free single-loop algorithms, namely PF-AGP-NSC algorithm, PF-AGP-NC algorithm and PF-AGP-NL algorithm, are proposed to solve the smooth nonconvex-strongly concave, nonconvex-concave minimax problems and nonconvex-linear minimax problems respectively using line search without requiring any prior knowledge about parameters such as the Lipschtiz constant $L$ or the strongly concave modulus $\mu$. Furthermore, we prove that the total number of gradient calls required to obtain an $\varepsilon$-stationary point for the PF-AGP-NSC algorithm, the PF-AGP-NC algorithm, and the PF-AGP-NL algorithm are upper bounded by $\mathcal{O}\left( L^2\kappa^3\varepsilon^{-2} \right)$, $\mathcal{O}\left( \log^2(L)L^4\varepsilon^{-4} \right)$, and $\mathcal{O}\left( L^3\varepsilon^{-3} \right)$, respectively, where $\kappa$ is the condition number. To the best of our knowledge, PF-AGP-NC and PF-AGP-NL are the first completely parameter-free algorithms for solving nonconvex-concave and nonconvex-linear minimax problems, respectively. PF-AGP-NSC is a completely parameter-free algorithm for solving nonconvex-strongly concave minimax problems, achieving the best known complexity with respect to $\varepsilon$. Numerical results demonstrate the efficiency of the three proposed algorithms.

翻译：由于极小极大问题在各种新兴应用中的重要性，求解极小极大问题的高效算法近来受到越来越多的关注。然而，许多现有算法需要预先知道问题参数才能达到最优迭代复杂度。本文提出了三种完全无需参数的单循环算法，即PF-AGP-NSC算法、PF-AGP-NC算法和PF-AGP-NL算法，分别用于求解光滑非凸-强凹、非凸-凹极小极大问题以及非凸-线性极小极大问题。这些算法采用线搜索技术，无需任何关于Lipschitz常数$L$或强凹模量$\mu$等参数的先验知识。此外，我们证明了PF-AGP-NSC算法、PF-AGP-NC算法和PF-AGP-NL算法获得$\varepsilon$-稳定点所需的总梯度调用次数上界分别为$\mathcal{O}\left( L^2\kappa^3\varepsilon^{-2} \right)$、$\mathcal{O}\left( \log^2(L)L^4\varepsilon^{-4} \right)$和$\mathcal{O}\left( L^3\varepsilon^{-3} \right)$，其中$\kappa$为条件数。据我们所知，PF-AGP-NC和PF-AGP-NL分别是首个完全无需参数求解非凸-凹和非凸-线性极小极大问题的算法。PF-AGP-NSC是首个完全无需参数求解非凸-强凹极小极大问题的算法，其在$\varepsilon$维度上达到了已知最优复杂度。数值实验结果验证了所提三种算法的有效性。