Symmetry and Generalisation in Machine Learning

This work is about understanding the impact of invariance and equivariance on generalisation in supervised learning. We use the perspective afforded by an averaging operator to show that for any predictor that is not equivariant, there is an equivariant predictor with strictly lower test risk on all regression problems where the equivariance is correctly specified. This constitutes a rigorous proof that symmetry, in the form of invariance or equivariance, is a useful inductive bias. We apply these ideas to equivariance and invariance in random design least squares and kernel ridge regression respectively. This allows us to specify the reduction in expected test risk in more concrete settings and express it in terms of properties of the group, the model and the data. Along the way, we give examples and additional results to demonstrate the utility of the averaging operator approach in analysing equivariant predictors. In addition, we adopt an alternative perspective and formalise the common intuition that learning with invariant models reduces to a problem in terms of orbit representatives. The formalism extends naturally to a similar intuition for equivariant models. We conclude by connecting the two perspectives and giving some ideas for future work.