Outlier-robust estimation of a sparse linear model using $\ell_1$-penalized Huber's $M$-estimator

We study the problem of estimating a $p$-dimensional $s$-sparse vector in a linear model with Gaussian design and additive noise. In the case where the labels are contaminated by at most $o$ adversarial outliers, we prove that the $\ell_1$-penalized Huber's $M$-estimator based on $n$ samples attains the optimal rate of convergence $(s/n)^{1/2} + (o/n)$, up to a logarithmic factor. This is proved when the proportion of contaminated samples goes to zero at least as fast as $1/\log(n)$, but we argue that constant fraction of outliers can be achieved by slightly more involved techniques.