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In this work, we consider the problem of a network of agents collectivelyminimizing a sum of convex functions. The agents in our setting can only accesstheir local objective functions and exchange information with their immediateneighbors. Motivated by applications where computation is imperfect, including,but not limited to, empirical risk minimization (ERM) and online learning, weassume that only noisy estimates of the local gradients are available. Totackle this problem, we adapt a class of Nested Distributed Gradient methods(NEAR-DGD) to the stochastic gradient setting. These methods have minimalstorage requirements, are communication aware and perform well in settingswhere gradient computation is costly, while communication is relativelyinexpensive. We investigate the convergence properties of our method understandard assumptions for stochastic gradients, i.e. unbiasedness and boundedvariance. Our analysis indicates that our method converges to a neighborhood ofthe optimal solution with a linear rate for local strongly convex functions andappropriate constant steplengths. We also show that distributed optimizationwith stochastic gradients achieves a noise reduction effect similar tomini-batching, which scales favorably with network size. Finally, we presentnumerical results to demonstrate the effectiveness of our method.

Nested Distributed Gradient Methods with Stochastic Computation Errors