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

Learning with a linear loss function. Excess risk and estimation bounds  for ERM, minmax MOM and their regularized versions. Applications to  robustness in sparse PCA