In this paper, we consider a class of nonconvex-nonconcave minimax problems, i.e., NC-PL minimax problems, whose objective functions satisfy the Polyak-\L ojasiewicz (PL) condition with respect to the inner variable. We propose a zeroth-order alternating gradient descent ascent (ZO-AGDA) algorithm and a zeroth-order variance reduced alternating gradient descent ascent (ZO-VRAGDA) algorithm for solving NC-PL minimax problem under the deterministic and the stochastic setting, respectively. The total number of function value queries to obtain an $\epsilon$-stationary point of ZO-AGDA and ZO-VRAGDA algorithm for solving NC-PL minimax problem is upper bounded by $\mathcal{O}(\varepsilon^{-2})$ and $\mathcal{O}(\varepsilon^{-3})$, respectively. To the best of our knowledge, they are the first two zeroth-order algorithms with the iteration complexity gurantee for solving NC-PL minimax problems.

翻译：本文研究一类非凸非凹极小极大问题，即NC-PL极小极大问题，其目标函数关于内部变量满足Polyak-Łojasiewicz（PL）条件。我们分别针对确定性和随机性场景，提出了零阶交替梯度下降上升算法（ZO-AGDA）和零阶方差缩减交替梯度下降上升算法（ZO-VRAGDA），用于求解NC-PL极小极大问题。在求解NC-PL极小极大问题时，ZO-AGDA和ZO-VRAGDA算法达到ε-稳定点所需的总函数值查询次数上界分别为𝒪(ε⁻²)和𝒪(ε⁻³)。据我们所知，这是首次提出的两种具有迭代复杂度保证的零阶算法，用于求解NC-PL极小极大问题。