Completely Parameter-Free Single-Loop Algorithms for Nonconvex-Concave Minimax Problems

Abstract

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 $μ$. 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 $\mathcal{O}\left( L^2κ^3\varepsilon^{-2} \right)$, $\mathcal{O}\left( \log^2(L)L^4\varepsilon^{-4} \right)$, and $\mathcal{O}\left( L^3\varepsilon^{-3} \right)$, respectively, where $κ$ 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.