Graph neural networks (GNNs) use graph convolutions to exploit network invariances and learn meaningful features from network data. However, on large-scale graphs convolutions incur in high computational cost, leading to scalability limitations. Leveraging the graphon -- the limit object of a graph -- in this paper we consider the problem of learning a graphon neural network (WNN) -- the limit object of a GNN -- by training GNNs on graphs sampled Bernoulli from the graphon. Under smoothness conditions, we show that: (i) the expected distance between the learning steps on the GNN and on the WNN decreases asymptotically with the size of the graph, and (ii) when training on a sequence of growing graphs, gradient descent follows the learning direction of the WNN. Inspired by these results, we propose a novel algorithm to learn GNNs on large-scale graphs that, starting from a moderate number of nodes, successively increases the size of the graph during training. This algorithm is benchmarked on both a recommendation system and a decentralized control problem where it is shown to retain comparable performance, to its large-scale counterpart, at a reduced computational cost.
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