Generalization Error of Invariant Classifiers

TL;DR

This paper analyzes how invariance in classifiers reduces generalization error by focusing on the base space complexity, with theoretical insights and experiments on MNIST and CIFAR-10.

Contribution

It provides a theoretical framework linking invariance to reduced complexity and generalization error, applicable to classifiers like CNNs.

Findings

Invariant classifiers have lower generalization error proportional to base space complexity.

Conditions on geometry ensure base space complexity is much smaller than input space.

Experimental results confirm theoretical predictions on MNIST and CIFAR-10.

Abstract

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.