A randomized algorithm for nonconvex minimization with inexact evaluations and complexity guarantees

We consider minimization of a smooth nonconvex function with inexact oracle access to gradient and Hessian (but not the function value) to achieve $(\epsilon_{g}, \epsilon_{H})$-approximate second-order optimality. A novel feature of our method is that if an approximate direction of negative curvature is chosen as the step, we choose its sense to be positive or negative with equal probability. We also use relative inexactness measures on gradient and Hessian and relax the coupling between the first- and second-order tolerances $\epsilon_{g}$ and $\epsilon_{H}$. Our convergence analysis includes both an expectation bound based on martingale analysis and a high-probability bound based on concentration inequalities. We apply our algorithm to empirical risk minimization problems and obtain gradient sample complexity.