Fast Rates for Regularized Objectives

We study convergence properties of empirical minimization of a stochastic strongly convex objective, where the stochastic component is linear. We show that the value attained by the empirical minimizer converges to the optimal value with rate 1/n. The result applies, in particular, to the SVM objective. Thus, we obtain a rate of 1/n on the convergence of the SVM objective (with fixed regularization parameter)to its infinite data limit. We demonstrate how this is essential for obtaining certain type of oracle inequalities for SVMs.