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In this paper, we consider the decentralized, stochastic nonconvexstrongly-concave (NCSC) minimax problem with nonsmooth regularization terms onboth primal and dual variables, wherein a network of $m$ computing agentscollaborate via peer-to-peer communications. We consider when the couplingfunction is in expectation or finite-sum form and the double regularizers areconvex functions, applied separately to the primal and dual variables. Ouralgorithmic framework introduces a Lagrangian multiplier to eliminate theconsensus constraint on the dual variable. Coupling this withvariance-reduction (VR) techniques, our proposed method, entitled VRLM, by asingle neighbor communication per iteration, is able to achieve an$\mathcal{O}(\kappa^3\varepsilon^{-3})$ sample complexity under the generalstochastic setting, with either a big-batch or small-batch VR option, where$\kappa$ is the condition number of the problem and $\varepsilon$ is thedesired solution accuracy. With a big-batch VR, we can additionally achieve$\mathcal{O}(\kappa^2\varepsilon^{-2})$ communication complexity. Under thespecial finite-sum setting, our method with a big-batch VR can achieve an$\mathcal{O}(n + \sqrt{n} \kappa^2\varepsilon^{-2})$ sample complexity and$\mathcal{O}(\kappa^2\varepsilon^{-2})$ communication complexity, where $n$ isthe number of components in the finite sum. All complexity results match thebest-known results achieved by a few existing methods for solving special casesof the problem we consider. To the best of our knowledge, this is the firstwork which provides convergence guarantees for NCSC minimax problems withgeneral convex nonsmooth regularizers applied to both the primal and dualvariables in the decentralized stochastic setting. Numerical experiments areconducted on two machine learning problems. Our code is downloadable fromhttps://github.com/RPI-OPT/VRLM.

Variance-reduced accelerated methods for decentralized stochastic double-regularized nonconvex strongly-concave minimax problems