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Rethinking Generalisation

In this paper, a new approach to computing the generalisation performance is presented that assumes the distribution of risks, $ρ(r)$, for a learning scenario is known. From this, the expected error of a learning machine using empirical risk minimisation is computed for both classification and regression problems. A critical quantity in determining the generalisation performance is the power-law behaviour of $ρ(r)$ around its minimum value---a quantity we call attunement. The distribution $ρ(r)$ is computed for the case of all Boolean functions and for the perceptron used in two different problem settings. Initially a simplified analysis is presented where an independence assumption about the losses is made. A more accurate analysis is carried out taking into account chance correlations in the training set. This leads to corrections in the typical behaviour that is observed.