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.

翻译：本文研究输出受对抗性污染时的鲁棒估计问题。Nguyen与Tran（2012）提出了一种用于鲁棒参数估计的扩展Lasso方法，并给出了估计误差的收敛速度。近期，Dalalyan与Thompson（2019）通过建立若干有效不等式，获得了比Nguyen与Tran（2012）更快的收敛速度。他们的研究重点在于：扩展Lasso的最小化问题可转化为带$L_1$惩罚项的Huber损失函数惩罚化问题。其关键区别在于Huber损失函数包含一个不同于传统方法的额外调节参数。本文提出了一种异于Dalalyan与Thompson（2019）的证明方法，最终获得了与其相同的收敛速度。我们证明的核心意义在于利用了Huber函数的若干特殊性质，这类技巧在以往的证明中尚未被使用。