Asymptotic Characterisation of Robust Empirical Risk Minimisation Performance in the Presence of Outliers
This provides theoretical insights into robust regression methods for statisticians and machine learning practitioners, though it is incremental as it extends existing asymptotic analysis to outlier-corrupted data.
The paper analyzes robust linear regression with outliers in high dimensions, deriving exact asymptotic performance for empirical risk minimization with different loss functions. It shows that optimally-regularized ERM is asymptotically consistent for generalization error but requires oracle knowledge for estimation error consistency, and identifies parameter regions where Huber and ℓ₂ losses perform identically.
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 $α=n/d$, 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 losses, 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.