  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, we propose a completely parameter-free alternating gradient projection
(PF-AGP) algorithm to solve the smooth nonconvex-(strongly) concave minimax
problems using a backtracking strategy, which does not require prior knowledge
of parameters such as the Lipschtiz constant $L$ or the strongly concave
constant $\mu$. The PF-AGP algorithm utilizes a parameter-free gradient
projection step to alternately update the outer and inner variables in each
iteration. We show that the total number of gradient calls of the PF-AGP
algorithm to obtain an $\varepsilon$-stationary point for nonconvex-strongly
concave minimax problems is upper bounded by $\mathcal{O}\left(
L\kappa^3\varepsilon^{-2} \right)$ where $\kappa$ is the condition number,
while the total number of gradient calls to obtain an $\varepsilon$-stationary
point for nonconvex-concave minimax problems is upper bounded by
$\mathcal{O}\left( L^4\varepsilon^{-4} \right)$. As far as we know, this is the
first completely parameter-free algorithm for solving nonconvex-strongly
concave minimax problems, and it is also the completely parameter-free
algorithm which achieves the best iteration complexity in single loop method
for solving nonconvex-concave minimax problems. Numerical results validate the
efficiency of the proposed PF-AGP algorithm.
