  This work considers the problem of binary classification: given training data
$x_1, \dots, x_n$ from a certain population, together with associated labels
$y_1,\dots, y_n \in \left\{0,1 \right\}$, determine the best label for an
element $x$ not among the training data. More specifically, this work considers
a variant of the regularized empirical risk functional which is defined
intrinsically to the observed data and does not depend on the underlying
population. Tools from modern analysis are used to obtain a concise proof of
asymptotic consistency as regularization parameters are taken to zero at rates
related to the size of the sample. These analytical tools give a new framework
for understanding overfitting and underfitting, and rigorously connect the
notion of overfitting with a loss of compactness.
