  In this paper, we study the problem of distributed multi-agent optimization
over a network, where each agent possesses a local cost function that is smooth
and strongly convex. The global objective is to find a common solution that
minimizes the average of all cost functions. Assuming agents only have access
to unbiased estimates of the gradients of their local cost functions, we
consider a distributed stochastic gradient tracking method (DSGT) and a
gossip-like stochastic gradient tracking method (GSGT). We show that, in
expectation, the iterates generated by each agent are attracted to a
neighborhood of the optimal solution, where they accumulate exponentially fast
(under a constant stepsize choice). Under DSGT, the limiting (expected) error
bounds on the distance of the iterates from the optimal solution decrease with
the network size $n$, which is a comparable performance to a centralized
stochastic gradient algorithm. Moreover, we show that when the network is
well-connected, GSGT incurs lower communication cost than DSGT while
maintaining a similar computational cost. Numerical example further
demonstrates the effectiveness of the proposed methods.
