  We consider learning methods based on the regularization of a convex
empirical risk by a squared Hilbertian norm, a setting that includes linear
predictors and non-linear predictors through positive-definite kernels. In
order to go beyond the generic analysis leading to convergence rates of the
excess risk as $O(1/\sqrt{n})$ from $n$ observations, we assume that the
individual losses are self-concordant, that is, their third-order derivatives
are bounded by their second-order derivatives. This setting includes
least-squares, as well as all generalized linear models such as logistic and
softmax regression. For this class of losses, we provide a bias-variance
decomposition and show that the assumptions commonly made in least-squares
regression, such as the source and capacity conditions, can be adapted to
obtain fast non-asymptotic rates of convergence by improving the bias terms,
the variance terms or both.
