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.

我们考虑了一类在没有中央服务器的分散环境中的非光滑强凸 - 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（inexact PDHG）算法，该算法允许通用梯度计算预言更新原始和对偶变量。我们首先研究了不精确 PDHG 与随机方差减少梯度（SVRG）预言的性能。我们的数值研究揭示出了 IPDHG 与 SVRG 预言的迭代过程中的初始保守进展现象。为了解决这个问题，我们提出了一个简单且有效的切换思想，即在更新的初始阶段使用广义随机梯度（GSG）计算预言来加快迭代过程，然后在适当的时机切换到 SVRG 预言。提出的算法被命名为带压缩的分散近端切换随机梯度方法（C-DPSSG），并被证明以线性速率收敛到一个 ε- 精确的鞍点解。除了提供高精度的解决方案外，我们的研究还揭示出利用 GSG 和 SVRG 预言的最佳收敛阶段使 C-DPSSG 非常适合获得低 / 中等准确度的解决方案，这对某些应用非常有用。对两个基准机器学习应用进行的数值实验显示了 C-DPSSG 的竞争性能，验证了我们的理论发现。