Gradient descent finds the cubic-regularized nonconvex Newton step

We consider the minimization of a nonconvex quadratic form regularized by a cubic term, which may exhibit saddle points and a suboptimal local minimum. Nonetheless, we prove that, under mild assumptions, gradient descent approximates the global minimum to within ε accuracy in O(ε⁻¹ log(1/ε)) steps for large ε and O(log(1/ε)) steps for small ε (compared to a condition number we define), with at most logarithmic dependence on the problem dimension. When we use gradient descent to approximate the cubic-regularized Newton step, our result implies a rate of convergence to second-order stationary points of general smooth nonconvex functions.