Outlier Robust and Sparse Estimation of Linear Regression Coefficients

We consider outlier-robust and sparse estimation of linear regression coefficients, when covariate vectors and noises are sampled, respectively, from an $\mathfrak{L}$-subGaussian distribution and a heavy-tailed distribution. Additionally, the covariate vectors and noises are contaminated by adversarial outliers. We deal with two cases: the covariance matrix of the covariates is known or unknown. Particularly, in the known case, our estimator can attain a nearly information theoretical optimal error bound, and our error bound is sharper than those of earlier studies dealing with similar situations. Our estimator analysis relies heavily on generic chaining to derive sharp error bounds.