本文提出了一种镜像梯度下降梯度上升算法来求解单边相对光滑的非凸-凹极小极大问题。在算法的每次迭代中, 我们采用镜像梯度下降步来更新相对光滑的变量, 采用梯度上升投影步来更新目标函数中光滑的变量。本文在理论上证明了算法收敛到$\varepsilon$-近似一阶稳定点的迭代复杂度是$\mathcal{O}\left( \varepsilon^{-4} \right)$。

In this paper, we propose a mirror descent gradient ascent algorithm to solve one side relatively smooth nonconvex-concave minimax optimization problems. At each iteration of the proposed algorithm, a mirror descent step is performed to update the relatively smooth variable, while a gradient ascent projection step is used to update the smooth variable alternately. We also prove that the iteration complexity of the proposed algorithm is $\mathcal{O}\left( \varepsilon ^{-4} \right)$ to achieve an $\varepsilon$-approximate first-order stationary point.