  Traditionally in regression one minimizes the number of fitting parameters or
uses smoothing/regularization to trade training (TE) and generalization error
(GE). Driving TE to zero by increasing fitting degrees of freedom (dof) is
expected to increase GE. However modern big-data approaches, including deep
nets, seem to over-parametrize and send TE to zero (data interpolation) without
impacting GE. Overparametrization has the benefit that global minima of the
empirical loss function proliferate and become easier to find. These phenomena
have drawn theoretical attention. Regression and classification algorithms have
been shown that interpolate data but also generalize optimally. An interesting
related phenomenon has been noted: the existence of non-monotonic risk curves,
with a peak in GE with increasing dof. It was suggested that this peak
separates a classical regime from a modern regime where over-parametrization
improves performance. Similar over-fitting peaks were reported previously
(statistical physics approach to learning) and attributed to increased fitting
model flexibility. We introduce a generative and fitting model pair
("Misparametrized Sparse Regression" or MiSpaR) and show that the overfitting
peak can be dissociated from the point at which the fitting function gains
enough dof's to match the data generative model and thus provides good
generalization. This complicates the interpretation of overfitting peaks as
separating a "classical" from a "modern" regime. Data interpolation itself
cannot guarantee good generalization: we need to study the interpolation with
different penalty terms. We present analytical formulae for GE curves for
MiSpaR with $l_2$ and $l_1$ penalties, in the interpolating limit
$\lambda\rightarrow 0$.These risk curves exhibit important differences and help
elucidate the underlying phenomena.

Machine Learning;Data Analysis, Statistics and Probability;Machine Learning