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 the (version 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.

翻译：本文研究可视为经典经验风险最小化之鲁棒类比算法的渐近性质。这些策略基于用均值的鲁棒代理（如均值中位数估计量的某种变体）替代通常的经验平均值。众所周知，在比经典对应算法所需假设弱得多的条件下，此类估计量的超额风险常以最优速率收敛到零。然而，关于估计量本身的渐近性质（例如鲁棒类比最大似然估计量是否具有渐近有效性）所知甚少。我们朝回答这些问题迈出了一步：对于一大类参数化问题，适当定义的鲁棒风险代理的最小化器，通常以与最小化通常经验风险所得估计量相同的速率收敛于真实风险的最小化器，并常具有相同的渐近方差。