  In this work, we consider the problem of a network of agents collectively
minimizing a sum of convex functions. The agents in our setting can only access
their local objective functions and exchange information with their immediate
neighbors. Motivated by applications where computation is imperfect, including,
but not limited to, empirical risk minimization (ERM) and online learning, we
assume that only noisy estimates of the local gradients are available. To
tackle this problem, we adapt a class of Nested Distributed Gradient methods
(NEAR-DGD) to the stochastic gradient setting. These methods have minimal
storage requirements, are communication aware and perform well in settings
where gradient computation is costly, while communication is relatively
inexpensive. We investigate the convergence properties of our method under
standard assumptions for stochastic gradients, i.e. unbiasedness and bounded
variance. Our analysis indicates that our method converges to a neighborhood of
the optimal solution with a linear rate for local strongly convex functions and
appropriate constant steplengths. We also show that distributed optimization
with stochastic gradients achieves a noise reduction effect similar to
mini-batching, which scales favorably with network size. Finally, we present
numerical results to demonstrate the effectiveness of our method.
