Don’t Just Blame Over-parametrization for Over-confidence: Theoretical Analysis of Calibration in Binary Classification

Yu Bai, Song Mei, Huan Wang, Caiming Xiong
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:566-576, 2021.

Abstract

Modern machine learning models with high accuracy are often miscalibrated—the predicted top probability does not reflect the actual accuracy, and tends to be \emph{over-confident}. It is commonly believed that such over-confidence is mainly due to \emph{over-parametrization}, in particular when the model is large enough to memorize the training data and maximize the confidence. In this paper, we show theoretically that over-parametrization is not the only reason for over-confidence. We prove that \emph{logistic regression is inherently over-confident}, in the realizable, under-parametrized setting where the data is generated from the logistic model, and the sample size is much larger than the number of parameters. Further, this over-confidence happens for general well-specified binary classification problems as long as the activation is symmetric and concave on the positive part. Perhaps surprisingly, we also show that over-confidence is not always the case—there exists another activation function (and a suitable loss function) under which the learned classifier is \emph{under-confident} at some probability values. Overall, our theory provides a precise characterization of calibration in realizable binary classification, which we verify on simulations and real data experiments.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-bai21c, title = {Don’t Just Blame Over-parametrization for Over-confidence: Theoretical Analysis of Calibration in Binary Classification}, author = {Bai, Yu and Mei, Song and Wang, Huan and Xiong, Caiming}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {566--576}, 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/bai21c/bai21c.pdf}, url = {https://proceedings.mlr.press/v139/bai21c.html}, abstract = {Modern machine learning models with high accuracy are often miscalibrated—the predicted top probability does not reflect the actual accuracy, and tends to be \emph{over-confident}. It is commonly believed that such over-confidence is mainly due to \emph{over-parametrization}, in particular when the model is large enough to memorize the training data and maximize the confidence. In this paper, we show theoretically that over-parametrization is not the only reason for over-confidence. We prove that \emph{logistic regression is inherently over-confident}, in the realizable, under-parametrized setting where the data is generated from the logistic model, and the sample size is much larger than the number of parameters. Further, this over-confidence happens for general well-specified binary classification problems as long as the activation is symmetric and concave on the positive part. Perhaps surprisingly, we also show that over-confidence is not always the case—there exists another activation function (and a suitable loss function) under which the learned classifier is \emph{under-confident} at some probability values. Overall, our theory provides a precise characterization of calibration in realizable binary classification, which we verify on simulations and real data experiments.} }
Endnote
%0 Conference Paper %T Don’t Just Blame Over-parametrization for Over-confidence: Theoretical Analysis of Calibration in Binary Classification %A Yu Bai %A Song Mei %A Huan Wang %A Caiming Xiong %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-bai21c %I PMLR %P 566--576 %U https://proceedings.mlr.press/v139/bai21c.html %V 139 %X Modern machine learning models with high accuracy are often miscalibrated—the predicted top probability does not reflect the actual accuracy, and tends to be \emph{over-confident}. It is commonly believed that such over-confidence is mainly due to \emph{over-parametrization}, in particular when the model is large enough to memorize the training data and maximize the confidence. In this paper, we show theoretically that over-parametrization is not the only reason for over-confidence. We prove that \emph{logistic regression is inherently over-confident}, in the realizable, under-parametrized setting where the data is generated from the logistic model, and the sample size is much larger than the number of parameters. Further, this over-confidence happens for general well-specified binary classification problems as long as the activation is symmetric and concave on the positive part. Perhaps surprisingly, we also show that over-confidence is not always the case—there exists another activation function (and a suitable loss function) under which the learned classifier is \emph{under-confident} at some probability values. Overall, our theory provides a precise characterization of calibration in realizable binary classification, which we verify on simulations and real data experiments.
APA
Bai, Y., Mei, S., Wang, H. & Xiong, C.. (2021). Don’t Just Blame Over-parametrization for Over-confidence: Theoretical Analysis of Calibration in Binary Classification. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:566-576 Available from https://proceedings.mlr.press/v139/bai21c.html.

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