Parametric high-dimensional regression requires regularization terms to get interpretable models. The respective estimators correspond to regularized M-functionals which are naturally highly nonlinear. Their Gâteaux derivative, i.e., their influence curve linearizes the asymptotic bias of the estimator, but only up to a remainder term which is not guaranteed to tend (sufficiently fast) to zero uniformly on suitable tangent sets without profound arguments. We fill this gap by studying, in a unified framework, under which conditions the M-functionals corresponding to convex penalties as regularization are compactly differentiable, so that the estimators admit an asymptotically linear expansion. This key ingredient allows influence curves to reasonably enter model diagnosis and enable a fast, valid update formula, just requiring an evaluation of the corresponding influence curve at new data points. Moreover, this paves the way for optimally-robust estimators, bounding the influence curves in a suitable way.

参数化高维回归需要正则项才能获得可解释的模型。各个估计量对应于自然高度非线性的正则化M函数。他们的Géteaux导数，即他们的影响曲线使估计量的渐近偏差线性化，但仅剩一个余项，不能保证在适当的切线集上没有充分的论据就趋于（足够快）均匀趋于零。我们通过在一个统一的框架中进行研究来填补这一空白，在这种情况下，与正则化凸凸惩罚相对应的M函数可以微分微分，从而使估计量可以渐近线性展开。通过此关键要素，影响曲线可以合理地进入模型诊断并启用快速有效的更新公式，仅需要评估新数据点上的相应影响曲线。此外，这为最优稳健的估计器铺平了道路，以合适的方式限制了影响曲线。