We consider a class of non-smooth strongly convex-strongly concave saddle point problems in a decentralized setting without a central server. To solve a consensus formulation of problems in this class, we develop an inexact primal dual hybrid gradient (inexact PDHG) procedure that allows generic gradient computation oracles to update the primal and dual variables. We first investigate the performance of inexact PDHG with stochastic variance reduction gradient (SVRG) oracle. Our numerical study uncovers a significant phenomenon of initial conservative progress of iterates of IPDHG with SVRG oracle. To tackle this, we develop a simple and effective switching idea, where a generalized stochastic gradient (GSG) computation oracle is employed to hasten the iterates' progress to a saddle point solution during the initial phase of updates, followed by a switch to the SVRG oracle at an appropriate juncture. The proposed algorithm is named Decentralized Proximal Switching Stochastic Gradient method with Compression (C-DPSSG), and is proven to converge to an $\epsilon$-accurate saddle point solution with linear rate. Apart from delivering highly accurate solutions, our study reveals that utilizing the best convergence phases of GSG and SVRG oracles makes C-DPSSG well suited for obtaining solutions of low/medium accuracy faster, useful for certain applications. Numerical experiments on two benchmark machine learning applications show C-DPSSG's competitive performance which validate our theoretical findings.

翻译：我们考虑一类在去中心化设置下（无中央服务器）的非光滑强凸-强凹鞍点问题。为了解决这类问题的共识形式，我们开发了一种不精确原始对偶混合梯度（不精确PDHG）方法，该方法允许使用通用梯度计算预言机来更新原始变量和对偶变量。我们首先研究了使用随机方差缩减梯度（SVRG）预言机的不精确PDHG的性能。数值研究发现了一个显著现象：IPDHG结合SVRG预言机在迭代初始阶段存在保守进展。为解决这一问题，我们提出了一种简单有效的切换策略，即在更新初始阶段使用广义随机梯度（GSG）计算预言机加速迭代向鞍点解的收敛，随后在适当节点切换至SVRG预言机。所提算法命名为带压缩的去中心化近端切换随机梯度方法（C-DPSSG），并证明其能以线性速率收敛到$\epsilon$精度的鞍点解。研究表明，除能提供高精度解之外，利用GSG和SVRG预言机的最佳收敛阶段使C-DPSSG特别适用于快速获取低/中精度解，这有益于某些应用场景。在两个基准机器学习应用上的数值实验展示了C-DPSSG的竞争性表现，验证了我们的理论发现。