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This paper proposes two distributed random reshuffling methods, namelyGradient Tracking with Random Reshuffling (GT-RR) and Exact Diffusion withRandom Reshuffling (ED-RR), to solve the distributed optimization problem overa connected network, where a set of agents aim to minimize the average of theirlocal cost functions. Both algorithms invoke random reshuffling (RR) update foreach agent, inherit favorable characteristics of RR for minimizing smoothnonconvex objective functions, and improve the performance of previousdistributed random reshuffling methods both theoretically and empirically.Specifically, both GT-RR and ED-RR achieve the convergence rate of$O(1/[(1-\lambda)^{1/3}m^{1/3}T^{2/3}])$ in driving the (minimum) expectedsquared norm of the gradient to zero, where $T$ denotes the number of epochs,$m$ is the sample size for each agent, and $1-\lambda$ represents the spectralgap of the mixing matrix. When the objective functions further satisfy thePolyak-{\L}ojasiewicz (PL) condition, we show GT-RR and ED-RR both achieve$O(1/[(1-\lambda)mT^2])$ convergence rate in terms of the averaged expecteddifferences between the agents' function values and the global minimum value.Notably, both results are comparable to the convergence rates of centralized RRmethods (up to constant factors depending on the network topology) andoutperform those of previous distributed random reshuffling algorithms.Moreover, we support the theoretical findings with a set of numericalexperiments.

Distributed Random Reshuffling Methods with Improved Convergence