  We obtain robust and computationally efficient estimators for learning
several linear models that achieve statistically optimal convergence rate under
minimal distributional assumptions. Concretely, we assume our data is drawn
from a $k$-hypercontractive distribution and an $\epsilon$-fraction is
adversarially corrupted. We then describe an estimator that converges to the
optimal least-squares minimizer for the true distribution at a rate
proportional to $\epsilon^{2-2/k}$, when the noise is independent of the
covariates. We note that no such estimator was known prior to our work, even
with access to unbounded computation. The rate we achieve is
information-theoretically optimal and thus we resolve the main open question in
Klivans, Kothari and Meka [COLT'18].
  Our key insight is to identify an analytic condition that serves as a
polynomial relaxation of independence of random variables. In particular, we
show that when the moments of the noise and covariates are
negatively-correlated, we obtain the same rate as independent noise. Further,
when the condition is not satisfied, we obtain a rate proportional to
$\epsilon^{2-4/k}$, and again match the information-theoretic lower bound. Our
central technical contribution is to algorithmically exploit independence of
random variables in the "sum-of-squares" framework by formulating it as the
aforementioned polynomial inequality.
