  In contrast to the empirical mean, the Median-of-Means (MoM) is an estimator
of the mean $\theta$ of a square integrable r.v. $Z$, around which accurate
nonasymptotic confidence bounds can be built, even when $Z$ does not exhibit a
sub-Gaussian tail behavior. Thanks to the high confidence it achieves on
heavy-tailed data, MoM has found various applications in machine learning,
where it is used to design training procedures that are not sensitive to
atypical observations. More recently, a new line of work is now trying to
characterize and leverage MoM's ability to deal with corrupted data. In this
context, the present work proposes a general study of MoM's concentration
properties under the contamination regime, that provides a clear understanding
of the impact of the outlier proportion and the number of blocks chosen. The
analysis is extended to (multisample) $U$-statistics, i.e. averages over tuples
of observations, that raise additional challenges due to the dependence
induced. Finally, we show that the latter bounds can be used in a
straightforward fashion to derive generalization guarantees for pairwise
learning in a contaminated setting, and propose an algorithm to compute
provably reliable decision functions.
