Minimax problems have recently attracted a lot of research interests. A few efforts have been made to solve decentralized nonconvex strongly-concave (NCSC) minimax-structured optimization; however, all of them focus on smooth problems with at most a constraint on the maximization variable. In this paper, we make the first attempt on solving composite NCSC minimax problems that can have convex nonsmooth terms on both minimization and maximization variables. Our algorithm is designed based on a novel reformulation of the decentralized minimax problem that introduces a multiplier to absorb the dual consensus constraint. The removal of dual consensus constraint enables the most aggressive (i.e., local maximization instead of a gradient ascent step) dual update that leads to the benefit of taking a larger primal stepsize and better complexity results. In addition, the decoupling of the nonsmoothness and consensus on the dual variable eases the analysis of a decentralized algorithm; thus our reformulation creates a new way for interested researchers to design new (and possibly more efficient) decentralized methods on solving NCSC minimax problems. We show a global convergence result of the proposed algorithm and an iteration complexity result to produce a (near) stationary point of the reformulation. Moreover, a relation is established between the (near) stationarities of the reformulation and the original formulation. With this relation, we show that when the dual regularizer is smooth, our algorithm can have lower complexity results (with reduced dependence on a condition number) than existing ones to produce a near-stationary point of the original formulation. Numerical experiments are conducted on a distributionally robust logistic regression to demonstrate the performance of the proposed algorithm.

翻译：极小极大问题近年来吸引了大量研究关注。已有若干工作致力于求解去中心化非凸强凹（NCSC）极小极大结构优化问题，然而这些工作均聚焦于至多在最大化变量上存在约束的光滑问题。本文率先尝试求解可同时在最小化和最大化变量上包含凸非光滑项的复合NCSC极小极大问题。我们的算法基于一种新颖的去中心化极小极大问题重构设计，该重构引入乘子以吸收对偶一致性约束。对偶一致性约束的移除使得能够采用最激进的对偶更新策略（即局部最大化而非梯度上升步骤），从而获得采用更大原始步长和更优复杂度结果的优势。此外，对偶变量非光滑性与一致性的解耦简化了去中心化算法的分析，因此我们的重构为研究者设计求解NCSC极小极大问题的新型（且可能更高效）去中心化方法开辟了新途径。我们证明了所提算法的全局收敛性结果，以及产生重构问题（近似）稳定点的迭代复杂度结果。同时建立了重构问题与原始问题（近似）稳定点之间的关联。基于该关联，我们表明当对偶正则化项光滑时，与现有方法相比，我们的算法能以更低的复杂度（降低对条件数的依赖）生成原始问题的近似稳定点。通过在分布鲁棒逻辑回归任务上开展数值实验，验证了所提算法的性能。