We study robust linear regression in high-dimension, when both the dimension
$d$ and the number of data points $n$ diverge with a fixed ratio $\alpha=n/d$,
and study a data model that includes outliers. We provide exact asymptotics for
the performances of the empirical risk minimisation (ERM) using
$\ell_2$-regularised $\ell_2$, $\ell_1$, and Huber loss, which are the standard
approach to such problems. We focus on two metrics for the performance: the
generalisation error to similar datasets with outliers, and the estimation
error of the original, unpolluted function. Our results are compared with the
information theoretic Bayes-optimal estimation bound. For the generalization
error, we find that optimally-regularised ERM is asymptotically consistent in
the large sample complexity limit if one perform a simple calibration, and
compute the rates of convergence. For the estimation error however, we show
that due to a norm calibration mismatch, the consistency of the estimator
requires an oracle estimate of the optimal norm, or the presence of a
cross-validation set not corrupted by the outliers. We examine in detail how
performance depends on the loss function and on the degree of outlier
corruption in the training set and identify a region of parameters where the
optimal performance of the Huber loss is identical to that of the $\ell_2$
loss, offering insights into the use cases of different loss functions.

本文研究高维度的鲁棒线性回归，包括离群值和使用标准损失函数的经验风险最小化（ERMs）方法。结果显示，在相似数据集上，经过最优正则化的 ERM 在大样本复杂性极限下是渐近一致的，但在评估误差方面，由于规范标定的失配，估计器的一致性要求完美计算最优规范的预估值或存在未受离群值污染的交叉验证集。不同的损失函数在最优性能的使用情况下提供了有关使用情况的见解。