Exact Gap between Generalization Error and Uniform Convergence in Random Feature Models

Zitong Yang, Yu Bai, Song Mei
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:11704-11715, 2021.

Abstract

Recent work showed that there could be a large gap between the classical uniform convergence bound and the actual test error of zero-training-error predictors (interpolators) such as deep neural networks. To better understand this gap, we study the uniform convergence in the nonlinear random feature model and perform a precise theoretical analysis on how uniform convergence depends on the sample size and the number of parameters. We derive and prove analytical expressions for three quantities in this model: 1) classical uniform convergence over norm balls, 2) uniform convergence over interpolators in the norm ball (recently proposed by \citet{zhou2021uniform}), and 3) the risk of minimum norm interpolator. We show that, in the setting where the classical uniform convergence bound is vacuous (diverges to $\infty$), uniform convergence over the interpolators still gives a non-trivial bound of the test error of interpolating solutions. We also showcase a different setting where classical uniform convergence bound is non-vacuous, but uniform convergence over interpolators can give an improved sample complexity guarantee. Our result provides a first exact comparison between the test errors and uniform convergence bounds for interpolators beyond simple linear models.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-yang21a, title = {Exact Gap between Generalization Error and Uniform Convergence in Random Feature Models}, author = {Yang, Zitong and Bai, Yu and Mei, Song}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {11704--11715}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/yang21a/yang21a.pdf}, url = {https://proceedings.mlr.press/v139/yang21a.html}, abstract = {Recent work showed that there could be a large gap between the classical uniform convergence bound and the actual test error of zero-training-error predictors (interpolators) such as deep neural networks. To better understand this gap, we study the uniform convergence in the nonlinear random feature model and perform a precise theoretical analysis on how uniform convergence depends on the sample size and the number of parameters. We derive and prove analytical expressions for three quantities in this model: 1) classical uniform convergence over norm balls, 2) uniform convergence over interpolators in the norm ball (recently proposed by \citet{zhou2021uniform}), and 3) the risk of minimum norm interpolator. We show that, in the setting where the classical uniform convergence bound is vacuous (diverges to $\infty$), uniform convergence over the interpolators still gives a non-trivial bound of the test error of interpolating solutions. We also showcase a different setting where classical uniform convergence bound is non-vacuous, but uniform convergence over interpolators can give an improved sample complexity guarantee. Our result provides a first exact comparison between the test errors and uniform convergence bounds for interpolators beyond simple linear models.} }
Endnote
%0 Conference Paper %T Exact Gap between Generalization Error and Uniform Convergence in Random Feature Models %A Zitong Yang %A Yu Bai %A Song Mei %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-yang21a %I PMLR %P 11704--11715 %U https://proceedings.mlr.press/v139/yang21a.html %V 139 %X Recent work showed that there could be a large gap between the classical uniform convergence bound and the actual test error of zero-training-error predictors (interpolators) such as deep neural networks. To better understand this gap, we study the uniform convergence in the nonlinear random feature model and perform a precise theoretical analysis on how uniform convergence depends on the sample size and the number of parameters. We derive and prove analytical expressions for three quantities in this model: 1) classical uniform convergence over norm balls, 2) uniform convergence over interpolators in the norm ball (recently proposed by \citet{zhou2021uniform}), and 3) the risk of minimum norm interpolator. We show that, in the setting where the classical uniform convergence bound is vacuous (diverges to $\infty$), uniform convergence over the interpolators still gives a non-trivial bound of the test error of interpolating solutions. We also showcase a different setting where classical uniform convergence bound is non-vacuous, but uniform convergence over interpolators can give an improved sample complexity guarantee. Our result provides a first exact comparison between the test errors and uniform convergence bounds for interpolators beyond simple linear models.
APA
Yang, Z., Bai, Y. & Mei, S.. (2021). Exact Gap between Generalization Error and Uniform Convergence in Random Feature Models. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:11704-11715 Available from https://proceedings.mlr.press/v139/yang21a.html.

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