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

Beyond Least-Squares: Fast Rates for Regularized Empirical Risk Minimization through Self-Concordance