  A fundamental principle of learning theory is that there is a trade-off
between the complexity of a prediction rule and its ability to generalize.
Modern machine learning models do not obey this paradigm: They produce an
accurate prediction even with a perfect fit to the training set. We investigate
over-parameterized linear regression models focusing on the minimum norm
solution: This is the solution with the minimal norm that attains a perfect fit
to the training set. We utilize the recently proposed predictive normalized
maximum likelihood (pNML) learner which is the min-max regret solution for the
distribution-free setting. We derive an upper bound of this min-max regret
which is associated with the prediction uncertainty. We show that if the test
sample lies mostly in a subspace spanned by the eigenvectors associated with
the large eigenvalues of the empirical correlation matrix of the training data,
the model generalizes despite its over-parameterized nature. We demonstrate the
use of the pNML regret as a point-wise learnability measure on synthetic data
and successfully observe the double-decent phenomenon of the over-parameterized
models on UCI datasets.
