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

Decentralized gradient descent maximization method for composite nonconvex strongly-concave minimax problems