  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.
