Replica theory for learning curves for Gaussian processes on random graphs
This provides theoretical insights for non-parametric inference in graph-based machine learning, though it is incremental as it applies existing replica methods to a specific setting.
The authors tackled the problem of predicting learning curves for Gaussian process regression on random graphs, showing that replica theory yields exact performance predictions in the large-graph limit, with results for both global and local normalisation of the kernel.
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 defined as the 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 noisy function values, we show that replica techniques can be used to obtain exact performance 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.