  Graph Convolutional Networks (GCNs) have achieved impressive empirical
advancement across a wide variety of semi-supervised node classification tasks.
Despite their great success, training GCNs on large graphs suffers from
computational and memory issues. A potential path to circumvent these obstacles
is sampling-based methods, where at each layer a subset of nodes is sampled.
Although recent studies have empirically demonstrated the effectiveness of
sampling-based methods, these works lack theoretical convergence guarantees
under realistic settings and cannot fully leverage the information of evolving
parameters during optimization. In this paper, we describe and analyze a
general doubly variance reduction schema that can accelerate any sampling
method under the memory budget. The motivating impetus for the proposed schema
is a careful analysis of the variance of sampling methods where it is shown
that the induced variance can be decomposed into node embedding approximation
variance (zeroth-order variance) during forward propagation and
layerwise-gradient variance (first-order variance) during backward propagation.
We theoretically analyze the convergence of the proposed schema and show that
it enjoys an $\mathcal{O}(1/T)$ convergence rate. We complement our theoretical
results by integrating the proposed schema in different sampling methods and
applying them to different large real-world graphs.
