  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.
