  We consider a commonly studied supervised classification of a synthetic
dataset whose labels are generated by feeding a one-layer neural network with
random iid inputs. We study the generalization performances of standard
classifiers in the high-dimensional regime where $\alpha=n/d$ is kept finite in
the limit of a high dimension $d$ and number of samples $n$. Our contribution
is three-fold: First, we prove a formula for the generalization error achieved
by $\ell_2$ regularized classifiers that minimize a convex loss. This formula
was first obtained by the heuristic replica method of statistical physics.
Secondly, focussing on commonly used loss functions and optimizing the $\ell_2$
regularization strength, we observe that while ridge regression performance is
poor, logistic and hinge regression are surprisingly able to approach the
Bayes-optimal generalization error extremely closely. As $\alpha \to \infty$
they lead to Bayes-optimal rates, a fact that does not follow from predictions
of margin-based generalization error bounds. Third, we design an optimal loss
and regularizer that provably leads to Bayes-optimal generalization error.
