  We present an accelerated gradient method for non-convex optimization
problems with Lipschitz continuous first and second derivatives. The method
requires time $O(\epsilon^{-7/4} \log(1/ \epsilon) )$ to find an
$\epsilon$-stationary point, meaning a point $x$ such that $\|\nabla f(x)\| \le
\epsilon$. The method improves upon the $O(\epsilon^{-2} )$ complexity of
gradient descent and provides the additional second-order guarantee that
$\nabla^2 f(x) \succeq -O(\epsilon^{1/2})I$ for the computed $x$. Furthermore,
our method is Hessian free, i.e. it only requires gradient computations, and is
therefore suitable for large scale applications.
