  We study the problem of high-dimensional robust mean estimation in the
presence of a constant fraction of adversarial outliers. A recent line of work
has provided sophisticated polynomial-time algorithms for this problem with
dimension-independent error guarantees for a range of natural distribution
families.
  In this work, we show that a natural non-convex formulation of the problem
can be solved directly by gradient descent. Our approach leverages a novel
structural lemma, roughly showing that any approximate stationary point of our
non-convex objective gives a near-optimal solution to the underlying robust
estimation task. Our work establishes an intriguing connection between
algorithmic high-dimensional robust statistics and non-convex optimization,
which may have broader applications to other robust estimation tasks.
