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) the regression covariate is a sequence with i.i.d. random vectors drawn from sub-Gaussian distribution with unknown mean and known identity covariance and (b) the absolute moment of the random noise is finite, our method attains a faster convergence rate than Diakonikolas, Kong and Stewart (2019) and Cherapanamjeri et al. (2020). Furthermore, our result is minimax optimal up to constant factor. When (a) the regression covariate is a sequence with i.i.d. random vectors drawn from heavy tailed distribution with unknown mean and bounded kurtosis and (b) the absolute moment of the random noise is finite, our method attains a convergence rate, which is minimax optimal up to constant factor.