Robust estimation with Lasso when outputs are adversarially contaminated

TL;DR

This paper proves the convergence rate of a robust Lasso estimator under adversarial output contamination, utilizing unique properties of the Huber loss function for the first time.

Contribution

The paper provides a novel proof technique for the convergence rate of the extended Lasso with Huber loss, differing from prior methods and highlighting the role of Huber's properties.

Findings

Achieves the same convergence rate as Dalalyan and Thompson (2019)

Introduces a new proof technique based on Huber function properties

Demonstrates robustness of Lasso under adversarial contamination

Abstract

We consider robust estimation when outputs are adversarially contaminated. Nguyen and Tran (2012) proposed an extended Lasso for robust parameter estimation and then they showed the convergence rate of the estimation error. Recently, Dalalyan and Thompson (2019) gave some useful inequalities and then they showed a faster convergence rate than Nguyen and Tran (2012). They focused on the fact that the minimization problem of the extended Lasso can become that of the penalized Huber loss function with $L_1$ penalty. The distinguishing point is that the Huber loss function includes an extra tuning parameter, which is different from the conventional method. We give the proof, which is different from Dalalyan and Thompson (2019) and then we give the same convergence rate as Dalalyan and Thompson (2019). The significance of our proof is to use some specific properties of the Huber function. Such techniques have not been used in the past proofs.