  We study the problem of high-dimensional linear regression in a robust model
where an $\epsilon$-fraction of the samples can be adversarially corrupted. We
focus on the fundamental setting where the covariates of the uncorrupted
samples are drawn from a Gaussian distribution $\mathcal{N}(0, \Sigma)$ on
$\mathbb{R}^d$. We give nearly tight upper bounds and computational lower
bounds for this problem. Specifically, our main contributions are as follows:
  For the case that the covariance matrix is known to be the identity, we give
a sample near-optimal and computationally efficient algorithm that outputs a
candidate hypothesis vector $\widehat{\beta}$ which approximates the unknown
regression vector $\beta$ within $\ell_2$-norm $O(\epsilon \log(1/\epsilon)
\sigma)$, where $\sigma$ is the standard deviation of the random observation
noise. An error of $\Omega (\epsilon \sigma)$ is information-theoretically
necessary, even with infinite sample size. Prior work gave an algorithm for
this problem with sample complexity $\tilde{\Omega}(d^2/\epsilon^2)$ whose
error guarantee scales with the $\ell_2$-norm of $\beta$.
  For the case of unknown covariance, we show that we can efficiently achieve
the same error guarantee as in the known covariance case using an additional
$\tilde{O}(d^2/\epsilon^2)$ unlabeled examples. On the other hand, an error of
$O(\epsilon \sigma)$ can be information-theoretically attained with
$O(d/\epsilon^2)$ samples. We prove a Statistical Query (SQ) lower bound
providing evidence that this quadratic tradeoff in the sample size is inherent.
More specifically, we show that any polynomial time SQ learning algorithm for
robust linear regression (in Huber's contamination model) with estimation
complexity $O(d^{2-c})$, where $c>0$ is an arbitrarily small constant, must
incur an error of $\Omega(\sqrt{\epsilon} \sigma)$.
