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

• URL: http://arxiv.org/abs/2102.07856v1
• Date: Mon, 15 Feb 2021 21:38:09 GMT
• Title: Don't Just Blame Over-parametrization for Over-confidence: Theoretical Analysis of Calibration in Binary Classification
• Authors: Yu Bai, Song Mei, Huan Wang, Caiming Xiong
• Abstract summary: We show theoretically that over-parametrization is not the only reason for over-confidence. We prove that logistic regression is inherently over-confident, in the realizable, under-parametrized setting. Perhaps surprisingly, we also show that over-confidence is not always the case.
• Score: 58.03725169462616
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 over-confident. It is commonly believed that such over-confidence is mainly due to 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 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 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.

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