Stochastic Gradient Methods with Compressed Communication for Decentralized Saddle Point Problems

Abstract: We develop two compression based stochastic gradient algorithms to solve a class of non-smooth strongly convex-strongly concave saddle-point problems in a decentralized setting (without a central server). Our first algorithm is a Restart-based Decentralized Proximal Stochastic Gradient method with Compression (C-RDPSG) for general stochastic settings. We provide rigorous theoretical guarantees of C-RDPSG with gradient computation complexity and communication complexity of order $$\mathcal{O}( (1+\delta)⁴ \frac{1}{L^{2}{\kappa_f^2}\kappa_g²} \frac{1}{\epsilon} )$$, to achieve an $\epsilon$-accurate saddle-point solution, where $\delta$ denotes the compression factor, $\kappa_f$ and $\kappa_g$ denote respectively the condition numbers of objective function and communication graph, and $L$ denotes the smoothness parameter of the smooth part of the objective function. Next, we present a Decentralized Proximal Stochastic Variance Reduced Gradient algorithm with Compression (C-DPSVRG) for finite sum setting which exhibits gradient computation complexity and communication complexity of order $$\mathcal{O} \left((1+\delta) \max {\kappa_f^2, \sqrt{\delta}\kappa^{2_f\kappa_g,\kappa_g} } \log\left(\frac{1}{\epsilon}\right) \right)$$. Extensive numerical experiments show competitive performance of the proposed algorithms and provide support to the theoretical results obtained.