In this paper, we study zeroth-order algorithms for nonconvex minimax problems with coupled linear constraints under the deterministic and stochastic settings, which have attracted wide attention in machine learning, signal processing and many other fields in recent years, e.g., adversarial attacks in resource allocation problems and network flow problems etc. We propose two single-loop algorithms, namely the zeroth-order primal-dual alternating projected gradient (ZO-PDAPG) algorithm and the zeroth-order regularized momentum primal-dual projected gradient algorithm (ZO-RMPDPG), for solving deterministic and stochastic nonconvex-(strongly) concave minimax problems with coupled linear constraints. The iteration complexity of the two proposed algorithms to obtain an $\varepsilon$-stationary point are proved to be $\mathcal{O}(\varepsilon ^{-2})$ (resp. $\mathcal{O}(\varepsilon ^{-4})$) for solving nonconvex-strongly concave (resp. nonconvex-concave) minimax problems with coupled linear constraints under deterministic settings and $\tilde{\mathcal{O}}(\varepsilon ^{-3})$ (resp. $\tilde{\mathcal{O}}(\varepsilon ^{-6.5})$) under stochastic settings respectively. To the best of our knowledge, they are the first two zeroth-order algorithms with iterative complexity guarantees for solving nonconvex-(strongly) concave minimax problems with coupled linear constraints under the deterministic and stochastic settings. The proposed ZO-RMPDPG algorithm, when specialized to stochastic nonconvex-concave minimax problems without coupled constraints, outperforms all existing zeroth-order algorithms by achieving a better iteration complexity, thus setting a new state-of-the-art.

翻译：本文研究了确定性与随机性设置下带耦合线性约束的非凸极小极大问题的零阶算法，此类问题近年来在机器学习、信号处理及众多其他领域（如资源分配问题和网络流问题中的对抗攻击等）引起了广泛关注。我们提出了两种单循环算法，即零阶原始对偶交替投影梯度（ZO-PDAPG）算法和零阶正则化动量原始对偶投影梯度（ZO-RMPDPG）算法，用于求解带耦合线性约束的确定性与随机性非凸-（强）凹极小极大问题。在确定性设置下，所提两种算法获得 $\varepsilon$-稳定点的迭代复杂度被证明分别为 $\mathcal{O}(\varepsilon ^{-2})$（对应非凸-强凹问题）和 $\mathcal{O}(\varepsilon ^{-4})$（对应非凸-凹问题）；在随机性设置下，则分别为 $\tilde{\mathcal{O}}(\varepsilon ^{-3})$ 和 $\tilde{\mathcal{O}}(\varepsilon ^{-6.5})$。据我们所知，这是首次在确定性与随机性设置下，为求解带耦合线性约束的非凸-（强）凹极小极大问题提供具有迭代复杂度保证的两种零阶算法。特别地，当所提出的ZO-RMPDPG算法专门用于求解无耦合约束的随机非凸-凹极小极大问题时，其以更优的迭代复杂度超越了所有现有零阶算法，从而确立了新的最优性能基准。