Asymptotic Normality of Robust Risk Minimizers

This paper investigates asymptotic properties of algorithms that can be viewed as robust analogues of the classical empirical risk minimization.These strategies are based on replacing the usual empirical average by a robust proxy of the mean, such as a variant of the median of means estimator.It is well known by now that the excess risk of resulting estimators often converges to zero at optimal rates under much weaker assumptions than those required by their classical counterparts.However, less is known about the asymptotic properties of the estimators themselves, for instance, whether robust analogues of the maximum likelihood estimators are asymptotically efficient.We make a step towards answering these questions and show that for a wide class of parametric problems, minimizers of the appropriately defined robust proxy of the risk converge to the minimizers of the true risk at the same rate, and often have the same asymptotic variance, as the estimators obtained by minimizing the usual empirical risk.Finally, we discuss the computational aspects of the problem and demonstrate the numerical performance of the methods under consideration in numerical experiments.