On the Variance, Admissibility, and Stability of Empirical Risk Minimization

It is well known that Empirical Risk Minimization (ERM) may attain minimax suboptimal rates in terms of the mean squared error (Birgé and Massart, 1993). In this paper, we prove that, under relatively mild assumptions, the suboptimality of ERM must be due to its bias. Namely, the variance error term of ERM (in terms of the bias and variance decomposition) enjoys the minimax rate. In the fixed design setting, we provide an elementary proof of this result using the probabilistic method. Then, we extend our proof to the random design setting for various models.