  We study the consensus decentralized optimization problem where the objective
function is the average of $n$ agents private non-convex cost functions;
moreover, the agents can only communicate to their neighbors on a given network
topology. The stochastic learning setting is considered in this paper where
each agent can only access a noisy estimate of its gradient. Many decentralized
methods can solve such problem including EXTRA, Exact-Diffusion/D$^2$, and
gradient-tracking. Unlike the famed DSGD algorithm, these methods have been
shown to be robust to the heterogeneity across the local cost functions.
However, the established convergence rates for these methods indicate that
their sensitivity to the network topology is worse than DSGD. Such theoretical
results imply that these methods can perform much worse than DSGD over sparse
networks, which, however, contradicts empirical experiments where DSGD is
observed to be more sensitive to the network topology.
  In this work, we study a general stochastic unified decentralized algorithm
(SUDA) that includes the above methods as special cases. We establish the
convergence of SUDA under both non-convex and the Polyak-Lojasiewicz condition
settings. Our results provide improved network topology dependent bounds for
these methods (such as Exact-Diffusion/D$^2$ and gradient-tracking) compared
with existing literature. Moreover, our results show that these methods are
often less sensitive to the network topology compared to DSGD, which agrees
with numerical experiments.
