  We carefully study how well minimizing convex surrogate loss functions,
corresponds to minimizing the misclassification error rate for the problem of
binary classification with linear predictors. In particular, we show that
amongst all convex surrogate losses, the hinge loss gives essentially the best
possible bound, of all convex loss functions, for the misclassification error
rate of the resulting linear predictor in terms of the best possible margin
error rate. We also provide lower bounds for specific convex surrogates that
show how different commonly used losses qualitatively differ from each other.
