Adversarial robust weighted Huber regression

We propose a novel method to estimate the coefficients of linear regression when outputs and inputs are contaminated by malicious outliers. Our method consists of two-step: (i) Make appropriate weights $\left\{\hat{w}_i\right\}_{i=1}^n$ such that the weighted sample mean of regression covariates robustly estimates the population mean of the regression covariate, (ii) Process Huber regression using $\left\{\hat{w}_i\right\}_{i=1}^n$. When (a-1) the regression covariate is a sequence with i.i.d. random vectors drawn from sub-Gaussian distribution satisfying $L_4$-$L_2$ norm equivalence with unknown mean and known identity covariance and (a-2) the absolute moment of the random noise is finite, our method attains a convergence rate, which is information theoretically optimal up to constant factor about noise term. When (b-1) the regression covariate is a sequence with i.i.d. random vectors drawn from heavy tailed distribution satisfying $L_4$-$L_2$ norm equivalence with unknown mean and (b-2) the absolute moment of the random noise is finite, our method attains a convergence rate, which is information theoretically optimal up to constant factor.