  This paper considers decentralized stochastic optimization over a network of
$n$ nodes, where each node possesses a smooth non-convex local cost function
and the goal of the networked nodes is to find an $\epsilon$-accurate
first-order stationary point of the sum of the local costs. We focus on an
online setting, where each node accesses its local cost only by means of a
stochastic first-order oracle that returns a noisy version of the exact
gradient. In this context, we propose a novel single-loop decentralized hybrid
variance-reduced stochastic gradient method, called GT-HSGD, that outperforms
the existing approaches in terms of both the oracle complexity and practical
implementation. The GT-HSGD algorithm implements specialized local hybrid
stochastic gradient estimators that are fused over the network to track the
global gradient. Remarkably, GT-HSGD achieves a network topology-independent
oracle complexity of $O(n^{-1}\epsilon^{-3})$ when the required error tolerance
$\epsilon$ is small enough, leading to a linear speedup with respect to the
centralized optimal online variance-reduced approaches that operate on a single
node. Numerical experiments are provided to illustrate our main technical
results.
