Article,
Asymptotic normality of robust risk minimizers
(2020)cite arxiv:2004.02328.
Abstract
This paper investigates asymptotic properties of a class 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 median-of-means estimator. It is well known by
now that the excess risk of resulting estimators often converges to 0 at the
optimal rates under much weaker assumptions than those required by their
"classical" counterparts. However, much less is known about 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 classical M-estimators.
Moreover, our results show that the algorithms based on "tournaments" or
"min-max" type procedures asymptotically outperform algorithms based on direct
risk minimization.
