Stochastic Gradient Methods with Compressed Communication for Decentralized Saddle Point Problems
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We propose two stochastic gradient algorithms to solve a class of saddle-point problems in a decentralized setting (without a central server). The proposed algorithms are the first to achieve sub-linear/linear computation and communication complexities using respectively stochastic gradient/stochastic variance reduced gradient oracles with compressed information exchange to solve non-smooth strongly-convex strongly-concave saddle-point problems in decentralized setting. 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 𝒪( (1+δ)^4 1/L^2κ_f^2κ_g^2 1/ϵ ), to achieve an ϵ-accurate saddle-point solution, where δ denotes the compression factor, κ_f and κ_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 𝒪((1+δ)κ_f^2 κ_g log(1/ϵ)). Extensive numerical experiments show competitive performance of the proposed algorithms and provide support to the theoretical results obtained.
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