  We obtain risk bounds for Empirical Risk Minimizers (ERM) and minmax
Median-Of-Means (MOM) estimators based on loss functions that are both
Lipschitz and convex. Results for the ERM are derived without assumptions on
the outputs and under subgaussian assumptions on the design and a new "local
Bernstein assumption" on the class of predictors. Similar results are shown for
minmax MOM estimators in a close setting where the design is only supposed to
satisfy moment assumptions, relaxing the Subgaussian hypothesis necessary for
ERM. The analysis of minmax MOM estimators is not based on the small ball
assumption (SBA) as it was the case in the first analysis of minmax MOM
estimators. In particular, the basic example of non parametric statistics where
the learning class is the linear span of localized bases, that does not satisfy
SBA can now be handled. Finally, minmax MOM estimators are analysed in a
setting where the local Bernstein condition is also dropped out. It is shown to
achieve an oracle inequality with exponentially large probability under minimal
assumptions insuring the existence of all objects.
