  We study the fundamental problem of high-dimensional mean estimation in a
robust model where a constant fraction of the samples are adversarially
corrupted. Recent work gave the first polynomial time algorithms for this
problem with dimension-independent error guarantees for several families of
structured distributions.
  In this work, we give the first nearly-linear time algorithms for
high-dimensional robust mean estimation. Specifically, we focus on
distributions with (i) known covariance and sub-gaussian tails, and (ii)
unknown bounded covariance. Given $N$ samples on $\mathbb{R}^d$, an
$\epsilon$-fraction of which may be arbitrarily corrupted, our algorithms run
in time $\tilde{O}(Nd) / \mathrm{poly}(\epsilon)$ and approximate the true mean
within the information-theoretically optimal error, up to constant factors.
Previous robust algorithms with comparable error guarantees have running times
$\tilde{\Omega}(N d^2)$, for $\epsilon = \Omega(1)$.
  Our algorithms rely on a natural family of SDPs parameterized by our current
guess $\nu$ for the unknown mean $\mu^\star$. We give a win-win analysis
establishing the following: either a near-optimal solution to the primal SDP
yields a good candidate for $\mu^\star$ -- independent of our current guess
$\nu$ -- or the dual SDP yields a new guess $\nu'$ whose distance from
$\mu^\star$ is smaller by a constant factor. We exploit the special structure
of the corresponding SDPs to show that they are approximately solvable in
nearly-linear time. Our approach is quite general, and we believe it can also
be applied to obtain nearly-linear time algorithms for other high-dimensional
robust learning problems.
