Generalization Error of Invariant Classifiers

Jure Sokolic, Raja Giryes, Guillermo Sapiro, Miguel Rodrigues
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:1094-1103, 2017.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v54-sokolic17a, title = {{Generalization Error of Invariant Classifiers}}, author = {Sokolic, Jure and Giryes, Raja and Sapiro, Guillermo and Rodrigues, Miguel}, booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics}, pages = {1094--1103}, year = {2017}, editor = {Singh, Aarti and Zhu, Jerry}, volume = {54}, series = {Proceedings of Machine Learning Research}, month = {20--22 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v54/sokolic17a/sokolic17a.pdf}, url = {https://proceedings.mlr.press/v54/sokolic17a.html}, 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.} }
Endnote
%0 Conference Paper %T Generalization Error of Invariant Classifiers %A Jure Sokolic %A Raja Giryes %A Guillermo Sapiro %A Miguel Rodrigues %B Proceedings of the 20th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2017 %E Aarti Singh %E Jerry Zhu %F pmlr-v54-sokolic17a %I PMLR %P 1094--1103 %U https://proceedings.mlr.press/v54/sokolic17a.html %V 54 %X 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.
APA
Sokolic, J., Giryes, R., Sapiro, G. & Rodrigues, M.. (2017). Generalization Error of Invariant Classifiers. Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 54:1094-1103 Available from https://proceedings.mlr.press/v54/sokolic17a.html.

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