  We study prediction and estimation problems using empirical risk
minimization, relative to a general convex loss function. We obtain sharp error
rates even when concentration is false or is very restricted, for example, in
heavy-tailed scenarios. Our results show that the error rate depends on two
parameters: one captures the intrinsic complexity of the class, and essentially
leads to the error rate in a noise-free (or realizable) problem; the other
measures interactions between class members the target and the loss, and is
dominant when the problem is far from realizable. We also explain how one may
deal with outliers by choosing the loss in a way that is calibrated to the
intrinsic complexity of the class and to the noise-level of the problem (the
latter is measured by the distance between the target and the class).
