Replica theory for learning curves for Gaussian processes on random graphs

TL;DR

This paper uses replica theory from statistical physics to derive exact learning curves for Gaussian process regression on large random graphs, considering both global and local kernel normalisation methods.

Contribution

It introduces a replica-based analytical framework to predict Gaussian process learning curves on random graphs with a focus on kernel normalisation.

Findings

Exact performance predictions for Gaussian process regression on large random graphs.

Comparison of global versus local kernel normalisation effects.

Insights into the impact of graph structure on learning efficiency.

Abstract

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.