Matthew Urry, Peter Sollich
Statistical physics approaches can be used to derive accurate predictions for the performance of inference methods learning from potentially noisy data, as quantified by the learning curve giving average error versus number of training examples. We analyse a challenging problem in the area of non-parametric inference where an effectively infinite number of parameters has to be learned, specifically Gaussian process regression. When the inputs are vertices on a random graph and the outputs the noisy function values being fitted, we show that replica techniques can be used to obtain exact predictions in the limit of large graphs. The covariance of the Gaussian process prior is defined by a random walk kernel, the discrete analogue of squared exponential kernels on continuous spaces. Conventionally this kernel is normalised only globally, so that the prior variance can differ between vertices; as a more principled alternative we consider local normalisation, where the prior variance is uniform. We compare our predictions with numerically simulated learning curves for two paradigmatic graph ensembles: Erdos-Renyi graphs with a Poisson degree distribution, and an ensemble with a power law degree distribution. We find excellent agreement between our predictions and numerical simulations. We also compare our predictions to existing learning curve approximations. These show significant deviations; we analyse briefly where these arise.
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http://arxiv.org/abs/1202.5918
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