Gradient Descent Efficiently Finds the Cubic-Regularized Non-Convex Newton Step

We consider the minimization of non-convex quadratic forms regularized by a cubic term, which exhibit multiple saddle points and poor local minima. Nonetheless, we prove that, under mild assumptions, gradient descent approximates the $\textit{global minimum}$ to within $\varepsilon$ accuracy in $O(\varepsilon^{-1}\log(1/\varepsilon))$ steps for large $\varepsilon$ and $O(\log(1/\varepsilon))$ steps for small $\varepsilon$ (compared to a condition number we define), with at most logarithmic dependence on the problem dimension. When we use gradient descent to approximate the Nesterov-Polyak cubic-regularized Newton step, our result implies a rate of convergence to second-order stationary points of general smooth non-convex functions.