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.

翻译：受若干实例的启发，本文考虑线性损失函数学习的一般框架。在此背景下，我们为四种估计量提供了以大概率成立的过剩风险与估计界：经验风险最小化（ERM）、极值中位数矩（minmax MOM）及其正则化形式。这些通用界被应用于稀疏主成分分析（PCA）的鲁棒性问题。特别地，我们改进了该问题现有的最优结果，在弱矩假设及对抗性污染数据下均获得了有效结论。