## AbstractWe study learning curves for Gaussian process regression which characterise performance in terms of the Bayes error averaged over datasets of a given size. Whilst learning curves are in general very difficult to calculate we show that for discrete input domains, where similarity between input points is characterized in terms nodes on a graph, accurate predictions can be obtained. These should in fact become exact for large graphs drawn from appropriate random graph ensembles. We focus on two types of ensemble. One is obtained by specifying (arbitrarily) the degree distribution and leads to sparse graphs, where each node is connected only to a finite number of others. The other is a community graph ensemble where we assume communities joined by a similar sparse superstructure. The calculation of the learning curves is based on translating the appropriate belief propagation equations to the graph ensemble. We demonstrate the accuracy of the predictions for Poisson (Erdos-Renyi) graphs and give some numerical results showing the need for a community orientated derivation of the learning curve.
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