  Motivated by several examples, we consider a general framework of learning
with linear loss functions. In this context, we provide excess risk and
estimation bounds that hold with large probability for four estimators: ERM,
minmax MOM and their regularized versions. These general bounds are applied for
the problem of robustness in sparse PCA. In particular, we improve the state of
the art result for this this problems, obtain results under weak moment
assumptions as well as for adversarial contaminated data.
