Variance-Reduced Decentralized Stochastic Optimization with Gradient Tracking--Part I: GT-SAGA

Abstract: In this paper, we study decentralized empirical risk minimization problems, where the goal is to minimize a finite-sum of smooth and strongly-convex functions available over a network of nodes. In this Part I, we propose \textbf{\texttt{GT-SAGA}}, a decentralized stochastic first-order algorithm based on gradient tracking \cite{DSGT_Pu,DSGT_Xin} and a variance-reduction technique called SAGA \cite{SAGA}. We develop the convergence analysis and the iteration complexity of this algorithm. We further demonstrate various trade-offs and discuss scenarios in which \textbf{\texttt{GT-SAGA}} achieves superior performance (in terms of the number of local gradient computations required) with respect to existing decentralized schemes. In Part II \cite{GT_SVRG} of this two-part paper, we develop and analyze \textbf{\texttt{GT-SVRG}}, a decentralized gradient tracking based implementation of SVRG \cite{SVRG}, another well-known variance-reduction technique.