## 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 characterised in terms of a graph, accurate predictions can be obtained. These should in fact become exact for large graphs drawn from a broad range of random graph ensembles with arbitrary degree distributions where each input (node) is connected only to a ﬁnite number of others. Our approach is based on translating the appropriate belief propagation equations to the graph ensemble. We demonstrate the accuracy of the predictions for Poisson (Erdos-Renyi) and regular random graphs, and discuss when and why previous approximations of the learning curve fail.
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