  Median-of-means (MOM) based procedures provide non-asymptotic and strong
deviation bounds even when data are heavy-tailed and/or corrupted. This work
proposes a new general way to bound the excess risk for MOM estimators. The
core technique is the use of VC-dimension (instead of Rademacher complexity) to
measure the statistical complexity. In particular, this allows to give the
first robust estimators for sparse estimation which achieves the so-called
subgaussian rate only assuming a finite second moment for the uncorrupted data.
By comparison, previous works using Rademacher complexities required a number
of finite moments that grows logarithmically with the dimension. With this
technique, we derive new robust sugaussian bounds for mean estimation in any
norm. We also derive a new robust estimator for covariance estimation that is
the first to achieve subgaussian bounds without $L_4-L_2$ norm equivalence.
