Accelerated methods for nonconvex optimization^∗

We present an accelerated gradient method for nonconvex optimization problems with Lipschitz continuous first and second derivatives. In a time O(⁻7/⁴ log(1/)), the method finds an -stationary point, meaning a point x such that ∇f(x) ≤ . The method improves upon the O(^{− 2}) complexity of gradient descent and provides the additional second-order guarantee that λ_min(∇²f(x)) −¹/² 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.