Covariate shift generalization, a typical case in out-of-distribution (OOD) generalization, requires a good performance on the unknown testing distribution, which varies from the accessible training distribution in the form of covariate shift. Recently, stable learning algorithms have shown empirical effectiveness to deal with covariate shift generalization on several learning models involving regression algorithms and deep neural networks. However, the theoretical explanations for such effectiveness are still missing. In this paper, we take a step further towards the theoretical analysis of stable learning algorithms by explaining them as feature selection processes. We first specify a set of variables, named minimal stable variable set, that is minimal and optimal to deal with covariate shift generalization for common loss functions, including the mean squared loss and binary cross entropy loss. Then we prove that under ideal conditions, stable learning algorithms could identify the variables in this set. Further analysis on asymptotic properties and error propagation are also provided. These theories shed light on why stable learning works for covariate shift generalization.

This paper studies the generalization error of invariant classifiers. In particular, we consider the common scenario where the classification task is invariant to certain transformations of the input, and that the classifier is constructed (or learned) to be invariant to these transformations. Our approach relies on factoring the input space into a product of a base space and a set of transformations. We show that whereas the generalization error of a non-invariant classifier is proportional to the complexity of the input space, the generalization error of an invariant classifier is proportional to the complexity of the base space. We also derive a set of sufficient conditions on the geometry of the base space and the set of transformations that ensure that the complexity of the base space is much smaller than the complexity of the input space. Our analysis applies to general classifiers such as convolutional neural networks. We demonstrate the implications of the developed theory for such classifiers with experiments on the MNIST and CIFAR-10 datasets.